Monty Hall problem
You are encouraged to solve this task according to the task description, using any language you may know.
Suppose you're on a game show and you're given the choice of three doors.
Behind one door is a car; behind the others, goats.
The car and the goats were placed randomly behind the doors before the show.
- Rules of the game
After you have chosen a door, the door remains closed for the time being.
The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it.
If both remaining doors have goats behind them, he chooses one randomly.
After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door.
Imagine that you chose Door 1 and the host opens Door 3, which has a goat.
He then asks you "Do you want to switch to Door Number 2?"
- The question
Is it to your advantage to change your choice?
- Note
The player may initially choose any of the three doors (not just Door 1), that the host opens a different door revealing a goat (not necessarily Door 3), and that he gives the player a second choice between the two remaining unopened doors.
- Task
Run random simulations of the Monty Hall game. Show the effects of a strategy of the contestant always keeping his first guess so it can be contrasted with the strategy of the contestant always switching his guess.
Simulate at least a thousand games using three doors for each strategy and show the results in such a way as to make it easy to compare the effects of each strategy.
- References
-
- Stefan Krauss, X. T. Wang, "The psychology of the Monty Hall problem: Discovering psychological mechanisms for solving a tenacious brain teaser.", Journal of Experimental Psychology: General, Vol 132(1), Mar 2003, 3-22 DOI: 10.1037/0096-3445.132.1.3
- A YouTube video: Monty Hall Problem - Numberphile.
11l
<lang 11l>V stay = 0 V sw = 0
L 1000
V lst = [1, 0, 0] random:shuffle(&lst) V ran = random:(3) V user = lst[ran] lst.pop(ran)
V huh = 0
L(i) lst
I i == 0
lst.pop(huh)
L.break
huh++
I user == 1
stay++
I lst[0] == 1
sw++
print(‘Stay = ’stay) print(‘Switch = ’sw)</lang>
8086 Assembly
<lang asm>time: equ 2Ch ; MS-DOS syscall to get current time puts: equ 9 ; MS-DOS syscall to print a string cpu 8086 bits 16 org 100h section .text ;;; Initialize the RNG with the current time mov ah,time int 21h mov di,cx ; RNG state is kept in DI and BP mov bp,dx mov dx,sw ; While switching doors, mov bl,1 call simrsl ; run simulations, mov dx,nsw ; While not switching doors, xor bl,bl ; run simulations. ;;; Print string in DX, run 65536 simulations (according to BL), ;;; then print the amount of cars won. simrsl: mov ah,puts ; Print the string int 21h xor cx,cx ; Run 65536 simulations call simul mov ax,si ; Print amount of cars mov bx,number ; String pointer mov cx,10 ; Divisor .dgt: xor dx,dx ; Divide AX by ten div cx add dl,'0' ; Add ASCII '0' to the remainder dec bx ; Move string pointer backwards mov [bx],dl ; Store digit in string test ax,ax ; If quotient not zero, jnz .dgt ; calculate next digit. mov dx,bx ; Print string starting at first digit mov ah,puts int 21h ret ;;; Run CX simulations. ;;; If BL = 0, don't switch doors, otherwise, always switch simul: xor si,si ; SI is the amount of cars won .loop: call door ; Behind which door is the car? xchg dl,al ; DL = car door call door ; Which door does the contestant choose? xchg ah,al ; AH = contestant door .monty: call door ; Which door does Monty open? cmp al,dl ; It can't be the door with the car, je .monty cmp al,ah ; or the door the contestant picked. je .monty test bl,bl ; Will the contestant switch doors? jz .nosw xor ah,al ; If so, he switches .nosw: cmp ah,dl ; Did he get the car? jne .next inc si ; If so, add a car .next: loop .loop ret ;;; Generate a pseudorandom byte in AL using "X ABC" method ;;; Use it to select a door (1,2,3). door: xchg bx,bp ; Load RNG state into byte-addressable xchg cx,di ; registers. .loop: inc bl ; X++ xor bh,ch ; A ^= C xor bh,bl ; A ^= X add cl,bh ; B += A mov al,cl ; C' = B shr al,1 ; C' >>= 1 add al,ch ; C' += C xor al,bh ; C' ^= A mov ch,al ; C = C' and al,3 ; ...but we only want the last two bits, jz .loop ; and if it was 0, get a new random number. xchg bx,bp ; Restore the registers xchg cx,di ret section .data sw: db 'When switching doors: $' nsw: db 'When not switching doors: $' db '*****' number: db 13,10,'$'</lang>
- Output:
When switching doors: 42841 When not switching doors: 22395
ActionScript
<lang actionscript>package { import flash.display.Sprite;
public class MontyHall extends Sprite { public function MontyHall() { var iterations:int = 30000; var switchWins:int = 0; var stayWins:int = 0;
for (var i:int = 0; i < iterations; i++) { var doors:Array = [0, 0, 0]; doors[Math.floor(Math.random() * 3)] = 1; var choice:int = Math.floor(Math.random() * 3); var shown:int;
do { shown = Math.floor(Math.random() * 3); } while (doors[shown] == 1 || shown == choice);
stayWins += doors[choice]; switchWins += doors[3 - choice - shown]; }
trace("Switching wins " + switchWins + " times. (" + (switchWins / iterations) * 100 + "%)"); trace("Staying wins " + stayWins + " times. (" + (stayWins / iterations) * 100 + "%)"); } } }</lang> Output:
Switching wins 18788 times. (62.626666666666665%) Staying wins 11212 times. (37.37333333333333%)
Ada
<lang ada>-- Monty Hall Game
with Ada.Text_Io; use Ada.Text_Io; with Ada.Float_Text_Io; use Ada.Float_Text_Io; with ada.Numerics.Discrete_Random;
procedure Monty_Stats is
Num_Iterations : Positive := 100000;
type Action_Type is (Stay, Switch);
type Prize_Type is (Goat, Pig, Car);
type Door_Index is range 1..3;
package Random_Prize is new Ada.Numerics.Discrete_Random(Door_Index);
use Random_Prize;
Seed : Generator;
Doors : array(Door_Index) of Prize_Type;
procedure Set_Prizes is
Prize_Index : Door_Index;
Booby_Prize : Prize_Type := Goat;
begin
Reset(Seed);
Prize_Index := Random(Seed);
Doors(Prize_Index) := Car;
for I in Doors'range loop
if I /= Prize_Index then
Doors(I) := Booby_Prize;
Booby_Prize := Prize_Type'Succ(Booby_Prize);
end if;
end loop;
end Set_Prizes;
function Play(Action : Action_Type) return Prize_Type is
Chosen : Door_Index := Random(Seed);
Monty : Door_Index;
begin
Set_Prizes;
for I in Doors'range loop
if I /= Chosen and Doors(I) /= Car then
Monty := I;
end if;
end loop;
if Action = Switch then
for I in Doors'range loop
if I /= Monty and I /= Chosen then
Chosen := I;
exit;
end if;
end loop;
end if;
return Doors(Chosen);
end Play;
Winners : Natural;
Pct : Float;
begin
Winners := 0;
for I in 1..Num_Iterations loop
if Play(Stay) = Car then
Winners := Winners + 1;
end if;
end loop;
Put("Stay : count" & Natural'Image(Winners) & " = ");
Pct := Float(Winners * 100) / Float(Num_Iterations);
Put(Item => Pct, Aft => 2, Exp => 0);
Put_Line("%");
Winners := 0;
for I in 1..Num_Iterations loop
if Play(Switch) = Car then
Winners := Winners + 1;
end if;
end loop;
Put("Switch : count" & Natural'Image(Winners) & " = ");
Pct := Float(Winners * 100) / Float(Num_Iterations);
Put(Item => Pct, Aft => 2, Exp => 0);
Put_Line("%");
end Monty_Stats;</lang> Results
Stay : count 34308 = 34.31% Switch : count 65695 = 65.69%
ALGOL 68
<lang algol68>INT trials=100 000;
PROC brand = (INT n)INT: 1 + ENTIER (n * random);
PROC percent = (REAL x)STRING: fixed(100.0*x/trials,0,2)+"%";
main: (
INT prize, choice, show, not shown, new choice;
INT stay winning:=0, change winning:=0, random winning:=0;
INT doors = 3;
[doors-1]INT other door;
TO trials DO
# put the prize somewhere #
prize := brand(doors);
# let the user choose a door #
choice := brand(doors);
# let us take a list of unchoosen doors #
INT k := LWB other door;
FOR j TO doors DO
IF j/=choice THEN other door[k] := j; k+:=1 FI
OD;
# Monty opens one... #
IF choice = prize THEN
# staying the user will win... Monty opens a random port#
show := other door[ brand(doors - 1) ];
not shown := other door[ (show+1) MOD (doors - 1 ) + 1]
ELSE # no random, Monty can open just one door... #
IF other door[1] = prize THEN
show := other door[2];
not shown := other door[1]
ELSE
show := other door[1];
not shown := other door[2]
FI
FI;
# the user randomly choose one of the two closed doors
(one is his/her previous choice, the second is the
one not shown ) #
other door[1] := choice;
other door[2] := not shown;
new choice := other door[ brand(doors - 1) ];
# now let us count if it takes it or not #
IF choice = prize THEN stay winning+:=1 FI;
IF not shown = prize THEN change winning+:=1 FI;
IF new choice = prize THEN random winning+:=1 FI
OD;
print(("Staying: ", percent(stay winning), new line ));
print(("Changing: ", percent(change winning), new line ));
print(("New random choice: ", percent(random winning), new line ))
)</lang> Sample output:
Staying: 33.62% Changing: 66.38% New random choice: 50.17%
APL
<lang apl> ∇ Run runs;doors;i;chosen;cars;goats;swap;stay;ix;prices [1] ⍝0: Monthy Hall problem [2] ⍝1: http://rosettacode.org/wiki/Monty_Hall_problem [3] [4] (⎕IO ⎕ML)←0 1 [5] prices←0 0 1 ⍝ 0=Goat, 1=Car [6] [7] ix←⊃,/{3?3}¨⍳runs ⍝ random indexes of doors (placement of car) [8] doors←(runs 3)⍴prices[ix] ⍝ matrix of doors [9] stay←+⌿doors[;?3] ⍝ chose randomly one door - is it a car? [10] swap←runs-stay ⍝ If not, then the other one is! [11] [12] ⎕←'Swap: ',(2⍕100×(swap÷runs)),'% its a car' [13] ⎕←'Stay: ',(2⍕100×(stay÷runs)),'% its a car'
∇</lang>
Run 100000 Swap: 66.54% it's a car Stay: 33.46% it's a car
Arturo
<lang rebol>stay: 0 swit: 0
loop 1..1000 'i [
lst: shuffle new [1 0 0]
rand: random 0 2
user: lst\[rand]
remove 'lst rand
huh: 0
loop lst 'i [
if zero? i [
remove 'lst huh
break
]
huh: huh + 1
]
if user=1 -> stay: stay+1 if and? [0 < size lst] [1 = first lst] -> swit: swit+1
]
print ["Stay:" stay] print ["Switch:" swit]</lang>
- Output:
Stay: 297 Switch: 549
AutoHotkey
<lang ahk>#SingleInstance, Force Iterations = 1000 Loop, %Iterations% {
If Monty_Hall(1)
Correct_Change++
Else
Incorrect_Change++
If Monty_Hall(2)
Correct_Random++
Else
Incorrect_Random++
If Monty_Hall(3)
Correct_Stay++
Else
Incorrect_Stay++
} Percent_Change := round(Correct_Change / Iterations * 100) Percent_Stay := round(Correct_Stay / Iterations * 100) Percent_Random := round(Correct_Random / Iterations * 100)
MsgBox,, Monty Hall Problem, These are the results:`r`n`r`nWhen I changed my guess, I got %Correct_Change% of %Iterations% (that's %Incorrect_Change% incorrect). That's %Percent_Change%`% correct.`r`n`r`nWhen I randomly changed my guess, I got %Correct_Random% of %Iterations% (that's %Incorrect_Random% incorrect). That's %Percent_Random%`% correct.`r`n`r`nWhen I stayed with my first guess, I got %Correct_Stay% of %Iterations% (that's %Incorrect_Stay% incorrect). That's %Percent_Stay%`% correct. ExitApp
Monty_Hall(Mode) ;Mode is 1 for change, 2 for random, or 3 for stay { Random, guess, 1, 3 Random, actual, 1, 3 Random, rand, 1, 2
show := guess = actual ? guess = 3 ? guess - rand : guess = 1 ? guess+rand : guess + 2*rand - 3 : 6 - guess - actual Mode := Mode = 2 ? 2*rand - 1: Mode Return, Mode = 1 ? 6 - guess - show = actual : guess = actual }</lang> Sample output:
These are the results: When I changed my guess, I got 659 of 1000 (that's 341 incorrect). That's 66% correct. When I randomly changed my guess, I got 505 of 1000 (that's 495 incorrect). That's 51% correct. When I stayed with my first guess, I got 329 of 1000 (that's 671 incorrect). That's 32% correct.
AWK
<lang awk>#!/bin/gawk -f
- Monty Hall problem
BEGIN { srand() doors = 3 iterations = 10000 # Behind a door: EMPTY = "empty"; PRIZE = "prize" # Algorithm used
KEEP = "keep"; SWITCH="switch"; RAND="random"; #
} function monty_hall( choice, algorithm ) {
# Set up doors
for ( i=0; i<doors; i++ ) {
door[i] = EMPTY }
# One door with prize
door[int(rand()*doors)] = PRIZE
chosen = door[choice] del door[choice]
#if you didn't choose the prize first time around then # that will be the alternative
alternative = (chosen == PRIZE) ? EMPTY : PRIZE
if( algorithm == KEEP) { return chosen } if( algorithm == SWITCH) { return alternative } return rand() <0.5 ? chosen : alternative }
function simulate(algo){ prizecount = 0 for(j=0; j< iterations; j++){ if( monty_hall( int(rand()*doors), algo) == PRIZE) { prizecount ++ } } printf " Algorithm %7s: prize count = %i, = %6.2f%%\n", \ algo, prizecount,prizecount*100/iterations }
BEGIN { print "\nMonty Hall problem simulation:" print doors, "doors,", iterations, "iterations.\n" simulate(KEEP) simulate(SWITCH) simulate(RAND)
}</lang> Sample output: <lang awk>bash$ ./monty_hall.awk
Monty Hall problem simulation: 3 doors, 10000 iterations.
Algorithm keep: prize count = 3411, = 34.11% Algorithm switch: prize count = 6655, = 66.55% Algorithm random: prize count = 4991, = 49.91%
bash$</lang>
BASIC
<lang qbasic>RANDOMIZE TIMER DIM doors(3) '0 is a goat, 1 is a car CLS switchWins = 0 stayWins = 0 FOR plays = 0 TO 32767 winner = INT(RND * 3) + 1 doors(winner) = 1'put a winner in a random door choice = INT(RND * 3) + 1'pick a door, any door DO shown = INT(RND * 3) + 1 'don't show the winner or the choice LOOP WHILE doors(shown) = 1 OR shown = choice stayWins = stayWins + doors(choice) 'if you won by staying, count it switchWins = switchWins + doors(3 - choice - shown) 'could have switched to win doors(winner) = 0 'clear the doors for the next test NEXT plays PRINT "Switching wins"; switchWins; "times." PRINT "Staying wins"; stayWins; "times."</lang> Output:
Switching wins 21805 times. Staying wins 10963 times.
BASIC256
<lang BASIC256> numTiradas = 1000000 permanece = 0 cambia = 0
for i = 1 to numTiradas pta_coche = int(rand * 3) + 1 pta_elegida = int(rand * 3) + 1 if pta_coche <> pta_elegida then pta_montys = 6 - pta_coche - pta_elegida else do pta_montys = int(Rand * 3) + 1 until pta_montys <> pta_coche end if # manteenr elección if pta_coche = pta_elegida then permanece += 1 # cambiar elección if pta_coche = 6 - pta_montys - pta_elegida then cambia +=1 next i
print "Si mantiene su elección, tiene un "; permanece / numTiradas * 100 ;"% de probabilidades de ganar." print "Si cambia, tiene un "; cambia / numTiradas * 100; "% de probabilidades de ganar." end </lang>
IS-BASIC
<lang IS-BASIC>100 PROGRAM "MontyH.bas" 110 RANDOMIZE 120 LET NUMGAMES=1000 130 LET CHANGING,NOTCHANGING=0 140 FOR I=0 TO NUMGAMES-1 150 LET PRIZEDOOR=RND(3)+1:LET CHOSENDOOR=RND(3)+1 160 IF CHOSENDOOR=PRIZEDOOR THEN 170 LET NOTCHANGING=NOTCHANGING+1 180 ELSE 190 LET CHANGING=CHANGING+1 200 END IF 210 NEXT 220 PRINT "Num of games:";NUMGAMES 230 PRINT "Wins not changing doors:";NOTCHANGING,NOTCHANGING/NUMGAMES*100;"% of total." 240 PRINT "Wins changing doors:",CHANGING,CHANGING/NUMGAMES*100;"% of total."</lang>
Sinclair ZX81 BASIC
Works with 1k of RAM.
This program could certainly be made more efficient. What is really going on, after all, is
if initial guess = car then
sticker wins
else
switcher wins;
but I take it that the point is to demonstrate the outcome to people who may not see that that's what is going on. I have therefore written the program in a deliberately naïve style, not assuming anything. <lang basic> 10 PRINT " WINS IF YOU"
20 PRINT "STICK","SWITCH" 30 LET STICK=0 40 LET SWITCH=0 50 FOR I=1 TO 1000 60 LET CAR=INT (RND*3) 70 LET GUESS=INT (RND*3) 80 LET SHOW=INT (RND*3) 90 IF SHOW=GUESS OR SHOW=CAR THEN GOTO 80
100 LET NEWGUESS=INT (RND*3) 110 IF NEWGUESS=GUESS OR NEWGUESS=SHOW THEN GOTO 100 120 IF GUESS=CAR THEN LET STICK=STICK+1 130 IF NEWGUESS=CAR THEN LET SWITCH=SWITCH+1 140 NEXT I 150 PRINT AT 2,0;STICK,SWITCH</lang>
- Output:
WINS IF YOU STICK SWITCH 341 659
True BASIC
<lang basic> LET numTiradas = 1000000
FOR i = 1 TO numTiradas
LET pta_coche = INT(RND * 3) + 1
LET pta_elegida = INT(RND * 3) + 1
IF pta_coche <> pta_elegida THEN
LET pta_montys = 6 - pta_coche - pta_elegida
ELSE
DO
LET pta_montys = INT(RND * 3) + 1
LOOP UNTIL pta_montys <> pta_coche
END IF
! mantener elección
IF pta_coche = pta_elegida THEN LET permanece = permanece + 1
! cambiar elección
IF pta_coche = 6 - pta_montys - pta_elegida THEN LET cambia = cambia + 1
NEXT i
PRINT "Cambiar gana el"; permanece / numTiradas * 100; "% de las veces." PRINT "Mantenerse gana el"; cambia / numTiradas * 100; "% de las veces." END </lang>
BBC BASIC
<lang bbcbasic> total% = 10000
FOR trial% = 1 TO total%
prize_door% = RND(3) : REM. The prize is behind this door
guess_door% = RND(3) : REM. The contestant guesses this door
IF prize_door% = guess_door% THEN
REM. The contestant guessed right, reveal either of the others
reveal_door% = RND(2)
IF prize_door% = 1 reveal_door% += 1
IF prize_door% = 2 AND reveal_door% = 2 reveal_door% = 3
ELSE
REM. The contestant guessed wrong, so reveal the non-prize door
reveal_door% = prize_door% EOR guess_door%
ENDIF
stick_door% = guess_door% : REM. The sticker doesn't change his mind
swap_door% = guess_door% EOR reveal_door% : REM. but the swapper does
IF stick_door% = prize_door% sticker% += 1
IF swap_door% = prize_door% swapper% += 1
NEXT trial%
PRINT "After a total of ";total%;" trials,"
PRINT "The 'sticker' won ";sticker%;" times (";INT(sticker%/total%*100);"%)"
PRINT "The 'swapper' won ";swapper%;" times (";INT(swapper%/total%*100);"%)"</lang>
Output:
After a total of 10000 trials, The 'sticker' won 3379 times (33%) The 'swapper' won 6621 times (66%)
C
<lang c>//Evidence of the Monty Hall solution.
- include <stdio.h>
- include <stdlib.h>
- include <time.h>
- define GAMES 3000000
int main(void){
unsigned i, j, k, choice, winsbyswitch=0, door[3];
srand(time(NULL)); //initialize random seed.
for(i=0; i<GAMES; i++){
door[0] = (!(rand()%2)) ? 1: 0; //give door 1 either a car or a goat randomly.
if(door[0]) door[1]=door[2]=0; //if 1st door has car, give other doors goats.
else{ door[1] = (!(rand()%2)) ? 1: 0; door[2] = (!door[1]) ? 1: 0; } //else, give 2nd door car or goat, give 3rd door what's left.
choice = rand()%3; //choose a random door.
//if the next door has a goat, and the following door has a car, or vice versa, you'd win if you switch.
if(((!(door[((choice+1)%3)])) && (door[((choice+2)%3)])) || (!(door[((choice+2)%3)]) && (door[((choice+1)%3)]))) winsbyswitch++;
}
printf("\nAfter %u games, I won %u by switching. That is %f%%. ", GAMES, winsbyswitch, (float)winsbyswitch*100.0/(float)i);
} </lang>
Output of one run:
After 3000000 games, I won 1999747 by switching. That is 66.658233%.
C#
<lang csharp>using System;
class Program {
static void Main(string[] args)
{
int switchWins = 0;
int stayWins = 0;
Random gen = new Random();
for(int plays = 0; plays < 1000000; plays++ )
{
int[] doors = {0,0,0};//0 is a goat, 1 is a car
var winner = gen.Next(3);
doors[winner] = 1; //put a winner in a random door
int choice = gen.Next(3); //pick a door, any door int shown; //the shown door do
{
shown = gen.Next(3); }
while (doors[shown] == 1 || shown == choice); //don't show the winner or the choice
stayWins += doors[choice]; //if you won by staying, count it
//the switched (last remaining) door is (3 - choice - shown), because 0+1+2=3
switchWins += doors[3 - choice - shown];
}
Console.Out.WriteLine("Staying wins " + stayWins + " times.");
Console.Out.WriteLine("Switching wins " + switchWins + " times.");
}
}</lang> Sample output:
Staying wins: 333830 Switching wins: 666170
C++
<lang cpp>#include <iostream>
- include <cstdlib>
- include <ctime>
int randint(int n) {
return (1.0*n*std::rand())/(1.0+RAND_MAX);
}
int other(int doorA, int doorB) {
int doorC;
if (doorA == doorB)
{
doorC = randint(2);
if (doorC >= doorA)
++doorC;
}
else
{
for (doorC = 0; doorC == doorA || doorC == doorB; ++doorC)
{
// empty
}
}
return doorC;
}
int check(int games, bool change) {
int win_count = 0;
for (int game = 0; game < games; ++game)
{
int const winning_door = randint(3);
int const original_choice = randint(3);
int open_door = other(original_choice, winning_door);
int const selected_door = change?
other(open_door, original_choice)
: original_choice;
if (selected_door == winning_door)
++win_count;
}
return win_count;
}
int main() {
std::srand(std::time(0));
int games = 10000; int wins_stay = check(games, false); int wins_change = check(games, true); std::cout << "staying: " << 100.0*wins_stay/games << "%, changing: " << 100.0*wins_change/games << "%\n";
}</lang> Sample output:
staying: 33.73%, changing: 66.9%
Chapel
<lang chapel>use Random;
param doors: int = 3; config const games: int = 1000;
config const maxTasks = 32; var numTasks = 1; while( games / numTasks > 1000000 && numTasks < maxTasks ) do numTasks += 1; const tasks = 1..#numTasks; const games_per_task = games / numTasks ; const remaining_games = games % numTasks ;
var wins_by_stay: [tasks] int;
coforall task in tasks {
var rand = new RandomStream();
for game in 1..#games_per_task {
var player_door = (rand.getNext() * 1000): int % doors ;
var winning_door = (rand.getNext() * 1000): int % doors ;
if player_door == winning_door then
wins_by_stay[ task ] += 1;
}
if task == tasks.last then {
for game in 1..#remaining_games {
var player_door = (rand.getNext() * 1000): int % doors ;
var winning_door = (rand.getNext() * 1000): int % doors ;
if player_door == winning_door then
wins_by_stay[ task ] += 1;
}
}
}
var total_by_stay = + reduce wins_by_stay;
var total_by_switch = games - total_by_stay; var percent_by_stay = ((total_by_stay: real) / games) * 100; var percent_by_switch = ((total_by_switch: real) / games) * 100;
writeln( "Wins by staying: ", total_by_stay, " or ", percent_by_stay, "%" ); writeln( "Wins by switching: ", total_by_switch, " or ", percent_by_switch, "%" ); if ( total_by_stay > total_by_switch ){
writeln( "Staying is the superior method." );
} else if( total_by_stay < total_by_switch ){
writeln( "Switching is the superior method." );
} else {
writeln( "Both methods are equal." );
} </lang> Sample output:
Wins by staying: 354 or 35.4% Wins by switching: 646 or 64.6% Switching is the superior method.
Clojure
<lang clojure>(ns monty-hall-problem
(:use [clojure.contrib.seq :only (shuffle)]))
(defn play-game [staying]
(let [doors (shuffle [:goat :goat :car])
choice (rand-int 3)
[a b] (filter #(not= choice %) (range 3))
alternative (if (= :goat (nth doors a)) b a)]
(= :car (nth doors (if staying choice alternative)))))
(defn simulate [staying times]
(let [wins (reduce (fn [counter _] (if (play-game staying) (inc counter) counter))
0
(range times))]
(str "wins " wins " times out of " times)))
</lang> <lang clojure>monty-hall-problem> (println "staying:" (simulate true 1000)) staying: wins 337 times out of 1000 nil monty-hall-problem> (println "switching:" (simulate false 1000)) switching: wins 638 times out of 1000 nil </lang>
COBOL
<lang cobol> IDENTIFICATION DIVISION.
PROGRAM-ID. monty-hall.
DATA DIVISION.
WORKING-STORAGE SECTION.
78 Num-Games VALUE 1000000.
*> These are needed so the values are passed to
*> get-rand-int correctly.
01 One PIC 9 VALUE 1.
01 Three PIC 9 VALUE 3.
01 doors-area.
03 doors PIC 9 OCCURS 3 TIMES.
01 choice PIC 9.
01 shown PIC 9.
01 winner PIC 9.
01 switch-wins PIC 9(7).
01 stay-wins PIC 9(7).
01 stay-wins-percent PIC Z9.99.
01 switch-wins-percent PIC Z9.99.
PROCEDURE DIVISION.
PERFORM Num-Games TIMES
MOVE 0 TO doors (winner)
CALL "get-rand-int" USING CONTENT One, Three,
REFERENCE winner
MOVE 1 TO doors (winner)
CALL "get-rand-int" USING CONTENT One, Three,
REFERENCE choice
PERFORM WITH TEST AFTER
UNTIL NOT(shown = winner OR choice)
CALL "get-rand-int" USING CONTENT One, Three,
REFERENCE shown
END-PERFORM
ADD doors (choice) TO stay-wins
ADD doors (6 - choice - shown) TO switch-wins
END-PERFORM
COMPUTE stay-wins-percent ROUNDED =
stay-wins / Num-Games * 100
COMPUTE switch-wins-percent ROUNDED =
switch-wins / Num-Games * 100
DISPLAY "Staying wins " stay-wins " times ("
stay-wins-percent "%)."
DISPLAY "Switching wins " switch-wins " times ("
switch-wins-percent "%)."
.
IDENTIFICATION DIVISION.
PROGRAM-ID. get-rand-int.
DATA DIVISION.
WORKING-STORAGE SECTION.
01 call-flag PIC X VALUE "Y".
88 first-call VALUE "Y", FALSE "N".
01 num-range PIC 9.
LINKAGE SECTION.
01 min-num PIC 9.
01 max-num PIC 9.
01 ret PIC 9.
PROCEDURE DIVISION USING min-num, max-num, ret.
*> Seed RANDOM once.
IF first-call
MOVE FUNCTION RANDOM(FUNCTION CURRENT-DATE (9:8))
TO num-range
SET first-call TO FALSE
END-IF
COMPUTE num-range = max-num - min-num + 1
COMPUTE ret =
FUNCTION MOD(FUNCTION RANDOM * 100000, num-range)
+ min-num
.
END PROGRAM get-rand-int.
END PROGRAM monty-hall.</lang>
- Output:
Staying wins 0333396 times (33.34%). Switching wins 0666604 times (66.66%).
ColdFusion
<lang cfm><cfscript>
function runmontyhall(num_tests) {
// number of wins when player switches after original selection
switch_wins = 0;
// number of wins when players "sticks" with original selection
stick_wins = 0;
// run all the tests
for(i=1;i<=num_tests;i++) {
// unconditioned potential for selection of each door
doors = [0,0,0];
// winning door is randomly assigned...
winner = randrange(1,3);
// ...and actualized in the array of real doors
doors[winner] = 1;
// player chooses one of three doors
choice = randrange(1,3);
do {
// monty randomly reveals a door...
shown = randrange(1,3);
}
// ...but monty only reveals empty doors;
// he will not reveal the door that the player has choosen
// nor will he reveal the winning door
while(shown==choice || doors[shown]==1);
// when the door the player originally selected is the winner, the "stick" option gains a point
stick_wins += doors[choice];
// to calculate the number of times the player would have won with a "switch", subtract the
// "value" of the chosen, "stuck-to" door from 1, the possible number of wins if the player
// chose and stuck with the winning door (1), the player would not have won by switching, so
// the value is 1-1=0 if the player chose and stuck with a losing door (0), the player would
// have won by switching, so the value is 1-0=1
switch_wins += 1-doors[choice];
}
// finally, simply run the percentages for each outcome
stick_percentage = (stick_wins/num_tests)*100;
switch_percentage = (switch_wins/num_tests)*100;
writeoutput('Number of Tests: ' & num_tests);
writeoutput('
Stick Wins: ' & stick_wins & ' ['& stick_percentage &'%]');
writeoutput('
Switch Wins: ' & switch_wins & ' ['& switch_percentage &'%]');
} runmontyhall(10000);
</cfscript></lang> Output:
Tests: 10,000 | Switching wins: 6655 [66.55%] | Sticking wins: 3345 [33.45%]
Common Lisp
<lang lisp>(defun make-round ()
(let ((array (make-array 3
:element-type 'bit
:initial-element 0)))
(setf (bit array (random 3)) 1)
array))
(defun show-goat (initial-choice array)
(loop for i = (random 3)
when (and (/= initial-choice i)
(zerop (bit array i)))
return i))
(defun won? (array i)
(= 1 (bit array i)))</lang>
<lang lisp>CL-USER> (progn (loop repeat #1=(expt 10 6)
for round = (make-round)
for initial = (random 3)
for goat = (show-goat initial round)
for choice = (loop for i = (random 3)
when (and (/= i initial)
(/= i goat))
return i)
when (won? round (random 3))
sum 1 into result-stay
when (won? round choice)
sum 1 into result-switch
finally (progn (format t "Stay: ~S%~%" (float (/ result-stay
#1# 1/100)))
(format t "Switch: ~S%~%" (float (/ result-switch
#1# 1/100))))))
Stay: 33.2716% Switch: 66.6593%</lang>
<lang lisp>
- Find out how often we win if we always switch
(defun rand-elt (s)
(elt s (random (length s))))
(defun monty ()
(let* ((doors '(0 1 2))
(prize (random 3));possible values: 0, 1, 2 (pick (random 3)) (opened (rand-elt (remove pick (remove prize doors))));monty opens a door which is not your pick and not the prize (other (car (remove pick (remove opened doors))))) ;you decide to switch to the one other door that is not your pick and not opened
(= prize other))) ; did you switch to the prize?
(defun monty-trials (n)
(count t (loop for x from 1 to n collect (monty))))
</lang>
D
<lang d>import std.stdio, std.random;
void main() {
int switchWins, stayWins;
while (switchWins + stayWins < 100_000) {
immutable carPos = uniform(0, 3); // Which door is car behind?
immutable pickPos = uniform(0, 3); // Contestant's initial pick.
int openPos; // Which door is opened by Monty Hall?
// Monty can't open the door you picked or the one with the car
// behind it.
do {
openPos = uniform(0, 3);
} while(openPos == pickPos || openPos == carPos);
int switchPos;
// Find position that's not currently picked by contestant and
// was not opened by Monty already.
for (; pickPos==switchPos || openPos==switchPos; switchPos++) {}
if (pickPos == carPos)
stayWins++;
else if (switchPos == carPos)
switchWins++;
else
assert(0); // Can't happen.
}
writefln("Switching/Staying wins: %d %d", switchWins, stayWins);
}</lang>
- Output:
Switching/Staying wins: 66609 33391
Dart
The class Game attempts to hide the implementation as much as possible, the play() function does not use any specifics of the implementation. <lang dart>int rand(int max) => (Math.random()*max).toInt();
class Game {
int _prize; int _open; int _chosen;
Game() {
_prize=rand(3);
_open=null;
_chosen=null;
}
void choose(int door) {
_chosen=door;
}
void reveal() {
if(_prize==_chosen) {
int toopen=rand(2);
if (toopen>=_prize)
toopen++;
_open=toopen;
} else {
for(int i=0;i<3;i++)
if(_prize!=i && _chosen!=i) {
_open=i;
break;
}
}
}
void change() {
for(int i=0;i<3;i++)
if(_chosen!=i && _open!=i) {
_chosen=i;
break;
}
}
bool hasWon() => _prize==_chosen;
String toString() {
String res="Prize is behind door $_prize";
if(_chosen!=null) res+=", player has chosen door $_chosen";
if(_open!=null) res+=", door $_open is open";
return res;
}
}
void play(int count, bool swap) {
int wins=0;
for(int i=0;i<count;i++) {
Game game=new Game();
game.choose(rand(3));
game.reveal();
if(swap)
game.change();
if(game.hasWon())
wins++;
}
String withWithout=swap?"with":"without";
double percent=(wins*100.0)/count;
print("playing $withWithout switching won $percent%");
}
test() {
for(int i=0;i<5;i++) {
Game g=new Game();
g.choose(i%3);
g.reveal();
print(g);
g.change();
print(g);
print("win==${g.hasWon()}");
}
}
main() {
play(10000,false); play(10000,true);
}</lang>
playing without switching won 33.32% playing with switching won 67.63%
Delphi
<lang Delphi>program MontyHall;
{$APPTYPE CONSOLE}
{$R *.res}
uses
System.SysUtils;
const
numGames = 1000000; // Number of games to run
var
switchWins, stayWins, plays: Int64; doors: array[0..2] of Integer; i, winner, choice, shown: Integer;
begin
switchWins := 0; stayWins := 0;
for plays := 1 to numGames do begin
//0 is a goat, 1 is a car
for i := 0 to 2 do
doors[i] := 0;
//put a winner in a random door winner := Random(3); doors[winner] := 1;
//pick a door, any door choice := Random(3);
//don't show the winner or the choice
repeat
shown := Random(3);
until (doors[shown] <> 1) and (shown <> choice);
//if you won by staying, count it stayWins := stayWins + doors[choice];
//the switched (last remaining) door is (3 - choice - shown), because 0+1+2=3 switchWins := switchWins + doors[3 - choice - shown]; end;
WriteLn('Staying wins ' + IntToStr(stayWins) + ' times.');
WriteLn('Switching wins ' + IntToStr(switchWins) + ' times.');
end. </lang>
- Output:
Staying wins 333253 times. Switching wins 666747 times.
Dyalect
<lang dyalect>var switchWins = 0 var stayWins = 0
for plays in 0..1000000 {
var doors = [0 ,0, 0] var winner = rnd(3) doors[winner] = 1 var choice = rnd(3) var shown = rnd(3)
while doors[shown] == 1 || shown == choice {
shown = rnd(3)
}
stayWins += doors[choice] switchWins += doors[3 - choice - shown]
}
print("Staying wins \(stayWins) times.") print("Switching wins \(switchWins) times.")</lang>
- Output:
Staying wins 431683 times. Switching wins 568318 times.
Eiffel
<lang eiffel> note description: "[ Monty Hall Problem as an Eiffel Solution
1. Set the stage: Randomly place car and two goats behind doors 1, 2 and 3. 2. Monty offers choice of doors --> Contestant will choose a random door or always one door. 2a. Door has Goat - door remains closed 2b. Door has Car - door remains closed 3. Monty offers cash --> Contestant takes or refuses cash. 3a. Takes cash: Contestant is Cash winner and door is revealed. Car Loser if car door revealed. 3b. Refuses cash: Leads to offer to switch doors. 4. Monty offers door switch --> Contestant chooses to stay or change. 5. Door reveal: Contestant refused cash and did or did not door switch. Either way: Reveal! 6. Winner and Loser based on door reveal of prize.
Car Winner: Chooses car door Cash Winner: Chooses cash over any door Goat Loser: Chooses goat door Car Loser: Chooses cash over car door or switches from car door to goat door ]" date: "$Date$" revision: "$Revision$"
class MH_APPLICATION
create make
feature {NONE} -- Initialization
make -- Initialize Current. do play_lets_make_a_deal ensure played_1000_games: game_count = times_to_play end
feature {NONE} -- Implementation: Access
live_contestant: attached like contestant -- Attached version of `contestant' do if attached contestant as al_contestant then Result := al_contestant else create Result check not_attached_contestant: False end end end
contestant: detachable TUPLE [first_door_choice, second_door_choice: like door_number_anchor; takes_cash, switches_door: BOOLEAN] -- Contestant for Current.
active_stage_door (a_door: like door_anchor): attached like door_anchor -- Attached version of `a_door'. do if attached a_door as al_door then Result := al_door else create Result check not_attached_door: False end end end
door_1, door_2, door_3: like door_anchor -- Doors with prize names and flags for goat and open (revealed).
feature {NONE} -- Implementation: Status
game_count, car_win_count, cash_win_count, car_loss_count, goat_loss_count, goat_avoidance_count: like counter_anchor switch_count, switch_win_count: like counter_anchor no_switch_count, no_switch_win_count: like counter_anchor -- Counts of games played, wins and losses based on car, cash or goat.
feature {NONE} -- Implementation: Basic Operations
prepare_stage -- Prepare the stage in terms of what doors have what prizes. do inspect new_random_of (3) when 1 then door_1 := door_with_car door_2 := door_with_goat door_3 := door_with_goat when 2 then door_1 := door_with_goat door_2 := door_with_car door_3 := door_with_goat when 3 then door_1 := door_with_goat door_2 := door_with_goat door_3 := door_with_car end active_stage_door (door_1).number := 1 active_stage_door (door_2).number := 2 active_stage_door (door_3).number := 3 ensure door_has_prize: not active_stage_door (door_1).is_goat or not active_stage_door (door_2).is_goat or not active_stage_door (door_3).is_goat consistent_door_numbers: active_stage_door (door_1).number = 1 and active_stage_door (door_2).number = 2 and active_stage_door (door_3).number = 3 end
door_number_having_prize: like door_number_anchor -- What door number has the car? do if not active_stage_door (door_1).is_goat then Result := 1 elseif not active_stage_door (door_2).is_goat then Result := 2 elseif not active_stage_door (door_3).is_goat then Result := 3 else check prize_not_set: False end end ensure one_to_three: between_1_and_x_inclusive (3, Result) end
door_with_car: attached like door_anchor -- Create a door with a car. do create Result Result.name := prize ensure not_empty: not Result.name.is_empty name_is_prize: Result.name.same_string (prize) end
door_with_goat: attached like door_anchor -- Create a door with a goat do create Result Result.name := gag_gift Result.is_goat := True ensure not_empty: not Result.name.is_empty name_is_prize: Result.name.same_string (gag_gift) is_gag_gift: Result.is_goat end
next_contestant: attached like live_contestant -- The next contestant on Let's Make a Deal! do create Result Result.first_door_choice := new_random_of (3) Result.second_door_choice := choose_another_door (Result.first_door_choice) Result.takes_cash := random_true_or_false if not Result.takes_cash then Result.switches_door := random_true_or_false end ensure choices_one_to_three: Result.first_door_choice <= 3 and Result.second_door_choice <= 3 switch_door_implies_no_cash_taken: Result.switches_door implies not Result.takes_cash end
choose_another_door (a_first_choice: like door_number_anchor): like door_number_anchor -- Make a choice from the remaining doors require one_to_three: between_1_and_x_inclusive (3, a_first_choice) do Result := new_random_of (3) from until Result /= a_first_choice loop Result := new_random_of (3) end ensure first_choice_not_second: a_first_choice /= Result result_one_to_three: between_1_and_x_inclusive (3, Result) end
play_lets_make_a_deal -- Play the game 1000 times local l_car_win, l_car_loss, l_cash_win, l_goat_loss, l_goat_avoided: BOOLEAN do from game_count := 0 invariant consistent_win_loss_counts: (game_count = (car_win_count + cash_win_count + goat_loss_count)) consistent_loss_avoidance_counts: (game_count = (car_loss_count + goat_avoidance_count)) until game_count >= times_to_play loop prepare_stage contestant := next_contestant l_cash_win := (live_contestant.takes_cash)
l_car_win := (not l_cash_win and (not live_contestant.switches_door and live_contestant.first_door_choice = door_number_having_prize) or (live_contestant.switches_door and live_contestant.second_door_choice = door_number_having_prize))
l_car_loss := (not live_contestant.switches_door and live_contestant.first_door_choice /= door_number_having_prize) or (live_contestant.switches_door and live_contestant.second_door_choice /= door_number_having_prize)
l_goat_loss := (not l_car_win and not l_cash_win)
l_goat_avoided := (not live_contestant.switches_door and live_contestant.first_door_choice = door_number_having_prize) or (live_contestant.switches_door and live_contestant.second_door_choice = door_number_having_prize)
check consistent_goats: l_goat_loss implies not l_goat_avoided end check consistent_car_win: l_car_win implies not l_car_loss and not l_cash_win and not l_goat_loss end check consistent_cash_win: l_cash_win implies not l_car_win and not l_goat_loss end check consistent_goat_avoidance: l_goat_avoided implies (l_car_win or l_cash_win) and not l_goat_loss end check consistent_car_loss: l_car_loss implies l_cash_win or l_goat_loss end
if l_car_win then car_win_count := car_win_count + 1 end if l_cash_win then cash_win_count := cash_win_count + 1 end if l_goat_loss then goat_loss_count := goat_loss_count + 1 end if l_car_loss then car_loss_count := car_loss_count + 1 end if l_goat_avoided then goat_avoidance_count := goat_avoidance_count + 1 end
if live_contestant.switches_door then switch_count := switch_count + 1 if l_car_win then switch_win_count := switch_win_count + 1 end else -- if not live_contestant.takes_cash and not live_contestant.switches_door then no_switch_count := no_switch_count + 1 if l_car_win or l_cash_win then no_switch_win_count := no_switch_win_count + 1 end end
game_count := game_count + 1
end
print ("%NCar Wins:%T%T " + car_win_count.out +
"%NCash Wins:%T%T " + cash_win_count.out +
"%NGoat Losses:%T%T " + goat_loss_count.out +
"%N-----------------------------" +
"%NTotal Win/Loss:%T%T" + (car_win_count + cash_win_count + goat_loss_count).out +
"%N%N" +
"%NCar Losses:%T%T " + car_loss_count.out +
"%NGoats Avoided:%T%T " + goat_avoidance_count.out +
"%N-----------------------------" +
"%NTotal Loss/Avoid:%T" + (car_loss_count + goat_avoidance_count).out +
"%N-----------------------------" +
"%NStaying Count/Win:%T" + no_switch_count.out + "/" + no_switch_win_count.out + " = " + (no_switch_win_count / no_switch_count * 100).out + " %%" +
"%NSwitch Count/Win:%T" + switch_count.out + "/" + switch_win_count.out + " = " + (switch_win_count / switch_count * 100).out + " %%"
)
end
feature {NONE} -- Implementation: Random Numbers
last_random: like random_number_anchor -- The last random number chosen.
random_true_or_false: BOOLEAN -- A randome True or False do Result := new_random_of (2) = 2 end
new_random_of (a_number: like random_number_anchor): like door_number_anchor -- A random number from 1 to `a_number'. do Result := (new_random \\ a_number + 1).as_natural_8 end
new_random: like random_number_anchor -- Random integer -- Each call returns another random number. do random_sequence.forth Result := random_sequence.item last_random := Result ensure old_random_not_new: old last_random /= last_random end
random_sequence: RANDOM -- Random sequence seeded from clock when called. attribute create Result.set_seed ((create {TIME}.make_now).milli_second) end
feature {NONE} -- Implementation: Constants
times_to_play: NATURAL_16 = 1000 -- Times to play the game.
prize: STRING = "Car" -- Name of the prize
gag_gift: STRING = "Goat" -- Name of the gag gift
door_anchor: detachable TUPLE [number: like door_number_anchor; name: STRING; is_goat, is_open: BOOLEAN] -- Type anchor for door tuples.
door_number_anchor: NATURAL_8 -- Type anchor for door numbers.
random_number_anchor: INTEGER -- Type anchor for random numbers.
counter_anchor: NATURAL_16 -- Type anchor for counters.
feature {NONE} -- Implementation: Contract Support
between_1_and_x_inclusive (a_number, a_value: like door_number_anchor): BOOLEAN -- Is `a_value' between 1 and `a_number'? do Result := (a_value > 0) and (a_value <= a_number) end
end </lang>
- Output:
Car Wins: 177 Cash Wins: 486 Goat Losses: 337 ----------------------------- Total Win/Loss: 1000 Car Losses: 657 Goats Avoided: 343 ----------------------------- Total Loss/Avoid: 1000 ----------------------------- Staying Count/Win: 742/573 = 77.223719676549862 % Switch Count/Win: 258/90 = 34.883720930232556 %
Elixir
<lang elixir>defmodule MontyHall do
def simulate(n) do
{stay, switch} = simulate(n, 0, 0)
:io.format "Staying wins ~w times (~.3f%)~n", [stay, 100 * stay / n]
:io.format "Switching wins ~w times (~.3f%)~n", [switch, 100 * switch / n]
end
defp simulate(0, stay, switch), do: {stay, switch}
defp simulate(n, stay, switch) do
doors = Enum.shuffle([:goat, :goat, :car])
guess = :rand.uniform(3) - 1
[choice] = [0,1,2] -- [guess, shown(doors, guess)]
if Enum.at(doors, choice) == :car, do: simulate(n-1, stay, switch+1),
else: simulate(n-1, stay+1, switch)
end
defp shown(doors, guess) do
[i, j] = Enum.shuffle([0,1,2] -- [guess])
if Enum.at(doors, i) == :goat, do: i, else: j
end
end
MontyHall.simulate(10000)</lang>
- Output:
Staying wins 3397 times (33.970%) Switching wins 6603 times (66.030%)
Emacs Lisp
<lang lisp>(defun montyhall (keep)
(let ((prize (random 3))
(choice (random 3)))
(if keep (= prize choice)
(/= prize choice))))
(let ((cnt 0))
(dotimes (i 10000) (and (montyhall t) (setq cnt (1+ cnt)))) (message "Strategy keep: %.3f%%" (/ cnt 100.0)))
(let ((cnt 0))
(dotimes (i 10000) (and (montyhall nil) (setq cnt (1+ cnt)))) (message "Strategy switch: %.3f%%" (/ cnt 100.0)))</lang>
- Output:
Strategy keep: 34.410% Strategy switch: 66.430%
Erlang
<lang erlang>-module(monty_hall).
-export([main/0]).
main() -> random:seed(now()), {WinStay, WinSwitch} = experiment(100000, 0, 0), io:format("Switching wins ~p times.\n", [WinSwitch]), io:format("Staying wins ~p times.\n", [WinStay]).
experiment(0, WinStay, WinSwitch) -> {WinStay, WinSwitch}; experiment(N, WinStay, WinSwitch) -> Doors = setelement(random:uniform(3), {0,0,0}, 1), SelectedDoor = random:uniform(3), OpenDoor = open_door(Doors, SelectedDoor), experiment( N - 1, WinStay + element(SelectedDoor, Doors), WinSwitch + element(6 - (SelectedDoor + OpenDoor), Doors) ).
open_door(Doors,SelectedDoor) -> OpenDoor = random:uniform(3), case (element(OpenDoor, Doors) =:= 1) or (OpenDoor =:= SelectedDoor) of true -> open_door(Doors, SelectedDoor); false -> OpenDoor end. </lang> Sample Output:
Switching wins 66595 times. Staying wins 33405 times.
Euphoria
<lang euphoria>integer switchWins, stayWins switchWins = 0 stayWins = 0
integer winner, choice, shown
for plays = 1 to 10000 do
winner = rand(3)
choice = rand(3)
while 1 do
shown = rand(3)
if shown != winner and shown != choice then
exit
end if
end while
stayWins += choice = winner
switchWins += 6-choice-shown = winner
end for
printf(1, "Switching wins %d times\n", switchWins)
printf(1, "Staying wins %d times\n", stayWins)
</lang>
Sample Output:
- Switching wins 6697 times
- Staying wins 3303 times
F#
I don't bother with having Monty "pick" a door, since you only win if you initially pick a loser in the switch strategy and you only win if you initially pick a winner in the stay strategy so there doesn't seem to be much sense in playing around the background having Monty "pick" doors. Makes it pretty simple to see why it's always good to switch. <lang fsharp>open System let monty nSims =
let rnd = new Random()
let SwitchGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
if winner <> pick then 1 else 0
let StayGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
if winner = pick then 1 else 0
let Wins (f:unit -> int) = seq {for i in [1..nSims] -> f()} |> Seq.sum
printfn "Stay: %d wins out of %d - Switch: %d wins out of %d" (Wins StayGame) nSims (Wins SwitchGame) nSims</lang>
Sample Output:
Stay: 332874 wins out of 1000000 - Switch: 667369 wins out of 1000000
I had a very polite suggestion that I simulate Monty's "pick" so I'm putting in a version that does that. I compare the outcome with my original outcome and, unsurprisingly, show that this is essentially a noop that has no bearing on the output, but I (kind of) get where the request is coming from so here's that version... <lang fsharp>let montySlower nSims =
let rnd = new Random()
let MontyPick winner pick =
if pick = winner then
[0..2] |> Seq.filter (fun i -> i <> pick) |> Seq.nth (rnd.Next(0,2))
else
3 - pick - winner
let SwitchGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
let monty = MontyPick winner pick
let pickFinal = 3 - monty - pick
// Show that Monty's pick has no effect...
if (winner <> pick) <> (pickFinal = winner) then
printfn "Monty's selection actually had an effect!"
if pickFinal = winner then 1 else 0
let StayGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
let monty = MontyPick winner pick
// This one's even more obvious than the above since pickFinal
// is precisely the same as pick
let pickFinal = pick
if (winner = pick) <> (winner = pickFinal) then
printfn "Monty's selection actually had an effect!"
if winner = pickFinal then 1 else 0
let Wins (f:unit -> int) = seq {for i in [1..nSims] -> f()} |> Seq.sum
printfn "Stay: %d wins out of %d - Switch: %d wins out of %d" (Wins StayGame) nSims (Wins SwitchGame) nSims</lang>
Forth
version 1
<lang forth>include random.fs
variable stay-wins variable switch-wins
- trial ( -- )
3 random 3 random ( prize choice ) = if 1 stay-wins +! else 1 switch-wins +! then ;
- trials ( n -- )
0 stay-wins ! 0 switch-wins ! dup 0 do trial loop cr stay-wins @ . [char] / emit dup . ." staying wins" cr switch-wins @ . [char] / emit . ." switching wins" ;
1000 trials</lang>
or in iForth:
<lang forth>0 value stay-wins 0 value switch-wins
- trial ( -- )
3 choose 3 choose ( -- prize choice ) = IF 1 +TO stay-wins exit ENDIF 1 +TO switch-wins ;
- trials ( n -- )
CLEAR stay-wins CLEAR switch-wins dup 0 ?DO trial LOOP CR stay-wins DEC. ." / " dup DEC. ." staying wins," CR switch-wins DEC. ." / " DEC. ." switching wins." ;</lang>
With output:
FORTH> 100000000 trials 33336877 / 100000000 staying wins, 66663123 / 100000000 switching wins. ok
version 2
While Forthers are known (and regarded) for always simplifying the problem, I think version 1 is missing the point here. The optimization can only be done if one already understands the game. For what it's worth, here is a simulation that takes all the turns of the game. <lang Forth>require random.fs here seed !
1000000 constant rounds variable wins variable car variable firstPick variable revealed defer applyStrategy
- isCar ( u - f ) car @ = ;
- remaining ( u u - u ) 3 swap - swap - ;
- setup 3 random car ! ;
- choose 3 random firstPick ! ;
- otherGoat ( - u ) car @ firstPick @ remaining ;
- randomGoat ( - u ) car @ 1+ 2 random + 3 mod ;
- reveal firstPick @ isCar IF randomGoat ELSE otherGoat THEN revealed ! ;
- keep ( - u ) firstPick @ ;
- switch ( - u ) firstPick @ revealed @ remaining ;
- open ( u - f ) isCar ;
- play ( - f ) setup choose reveal applyStrategy open ;
- record ( f ) 1 and wins +! ;
- run 0 wins ! rounds 0 ?DO play record LOOP ;
- result wins @ 0 d>f rounds 0 d>f f/ 100e f* ;
- .result result f. '%' emit ;
' keep IS applyStrategy run ." Keep door => " .result cr ' switch IS applyStrategy run ." Switch door => " .result cr bye</lang>
- Output:
Keep door => 33.2922 % Switch door => 66.7207 %
Fortran
<lang fortran>PROGRAM MONTYHALL
IMPLICIT NONE
INTEGER, PARAMETER :: trials = 10000 INTEGER :: i, choice, prize, remaining, show, staycount = 0, switchcount = 0 LOGICAL :: door(3) REAL :: rnum
CALL RANDOM_SEED
DO i = 1, trials
door = .FALSE.
CALL RANDOM_NUMBER(rnum)
prize = INT(3*rnum) + 1
door(prize) = .TRUE. ! place car behind random door
CALL RANDOM_NUMBER(rnum)
choice = INT(3*rnum) + 1 ! choose a door
DO
CALL RANDOM_NUMBER(rnum)
show = INT(3*rnum) + 1
IF (show /= choice .AND. show /= prize) EXIT ! Reveal a goat
END DO
SELECT CASE(choice+show) ! Calculate remaining door index
CASE(3)
remaining = 3
CASE(4)
remaining = 2
CASE(5)
remaining = 1
END SELECT
IF (door(choice)) THEN ! You win by staying with your original choice
staycount = staycount + 1
ELSE IF (door(remaining)) THEN ! You win by switching to other door
switchcount = switchcount + 1
END IF
END DO
WRITE(*, "(A,F6.2,A)") "Chance of winning by not switching is", real(staycount)/trials*100, "%" WRITE(*, "(A,F6.2,A)") "Chance of winning by switching is", real(switchcount)/trials*100, "%"
END PROGRAM MONTYHALL</lang> Sample Output
Chance of winning by not switching is 32.82% Chance of winning by switching is 67.18%
FreeBASIC
<lang freebasic>' version 19-01-2019 ' compile with: fbc -s console
Const As Integer max = 1000000 Randomize Timer
Dim As UInteger i, car_door, chosen_door, montys_door, stay, switch
For i = 1 To max
car_door = Fix(Rnd * 3) + 1
chosen_door = Fix(Rnd * 3) + 1
If car_door <> chosen_door Then
montys_door = 6 - car_door - chosen_door
Else
Do
montys_door = Fix(Rnd * 3) + 1
Loop Until montys_door <> car_door
End If
'Print car_door,chosen_door,montys_door
' stay
If car_door = chosen_door Then stay += 1
' switch
If car_door = 6 - montys_door - chosen_door Then switch +=1
Next
Print Using "If you stick to your choice, you have a ##.## percent" _
+ " chance to win"; stay / max * 100
Print Using "If you switched, you have a ##.## percent chance to win"; _
switch / max * 100
' empty keyboard buffer While InKey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
If you stick to your choice, you have a 33.32 percent chance to win If you switched, you have a 66.68 percent chance to win
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website, However they run on execution servers. By default remote servers are used, but they are limited in memory and processing power, since they are intended for demonstration and casual use. A local server can be downloaded and installed, it has no limitations (it runs in your own computer). Because of that, example programs can be fully visualized and edited, but some of them will not run if they require a moderate or heavy computation/memory resources, and no local server is being used.
In this page you can see the program(s) related to this task and their results.
Go
<lang go>package main
import ( "fmt" "math/rand" "time" )
func main() { games := 100000 r := rand.New(rand.NewSource(time.Now().UnixNano()))
var switcherWins, keeperWins, shown int for i := 0; i < games; i++ { doors := []int{0, 0, 0} doors[r.Intn(3)] = 1 // Set which one has the car choice := r.Intn(3) // Choose a door for shown = r.Intn(3); shown == choice || doors[shown] == 1; shown = r.Intn(3) {} switcherWins += doors[3 - choice - shown] keeperWins += doors[choice] } floatGames := float32(games) fmt.Printf("Switcher Wins: %d (%3.2f%%)\n", switcherWins, (float32(switcherWins) / floatGames * 100)) fmt.Printf("Keeper Wins: %d (%3.2f%%)", keeperWins, (float32(keeperWins) / floatGames * 100)) }</lang> Output:
Switcher Wins: 66542 (66.54%) Keeper Wins: 33458 (33.46%)
Haskell
<lang haskell>import System.Random (StdGen, getStdGen, randomR)
trials :: Int trials = 10000
data Door = Car | Goat deriving Eq
play :: Bool -> StdGen -> (Door, StdGen) play switch g = (prize, new_g)
where (n, new_g) = randomR (0, 2) g
d1 = [Car, Goat, Goat] !! n
prize = case switch of
False -> d1
True -> case d1 of
Car -> Goat
Goat -> Car
cars :: Int -> Bool -> StdGen -> (Int, StdGen) cars n switch g = f n (0, g)
where f 0 (cs, g) = (cs, g)
f n (cs, g) = f (n - 1) (cs + result, new_g)
where result = case prize of Car -> 1; Goat -> 0
(prize, new_g) = play switch g
main = do
g <- getStdGen
let (switch, g2) = cars trials True g
(stay, _) = cars trials False g2
putStrLn $ msg "switch" switch
putStrLn $ msg "stay" stay
where msg strat n = "The " ++ strat ++ " strategy succeeds " ++
percent n ++ "% of the time."
percent n = show $ round $
100 * (fromIntegral n) / (fromIntegral trials)</lang>
With a State monad, we can avoid having to explicitly pass around the StdGen so often. play and cars can be rewritten as follows:
<lang haskell>import Control.Monad.State
play :: Bool -> State StdGen Door play switch = do
i <- rand
let d1 = [Car, Goat, Goat] !! i
return $ case switch of
False -> d1
True -> case d1 of
Car -> Goat
Goat -> Car
where rand = do
g <- get
let (v, new_g) = randomR (0, 2) g
put new_g
return v
cars :: Int -> Bool -> StdGen -> (Int, StdGen) cars n switch g = (numcars, new_g)
where numcars = length $ filter (== Car) prize_list
(prize_list, new_g) = runState (replicateM n (play switch)) g</lang>
Sample output (for either implementation): <lang haskell>The switch strategy succeeds 67% of the time. The stay strategy succeeds 34% of the time.</lang>
HicEst
<lang hicest>REAL :: ndoors=3, doors(ndoors), plays=1E4
DLG(NameEdit = plays, DNum=1, Button='Go')
switchWins = 0 stayWins = 0
DO play = 1, plays
doors = 0 ! clear the doors winner = 1 + INT(RAN(ndoors)) ! door that has the prize doors(winner) = 1 guess = 1 + INT(RAN(doors)) ! player chooses his door
IF( guess == winner ) THEN ! Monty decides which door to open:
show = 1 + INT(RAN(2)) ! select 1st or 2nd goat-door
checked = 0
DO check = 1, ndoors
checked = checked + (doors(check) == 0)
IF(checked == show) open = check
ENDDO
ELSE
open = (1+2+3) - winner - guess
ENDIF
new_guess_if_switch = (1+2+3) - guess - open
stayWins = stayWins + doors(guess) ! count if guess was correct switchWins = switchWins + doors(new_guess_if_switch)
ENDDO
WRITE(ClipBoard, Name) plays, switchWins, stayWins
END</lang> <lang hicest>! plays=1E3; switchWins=695; stayWins=305; ! plays=1E4; switchWins=6673; stayWins=3327; ! plays=1E5; switchWins=66811; stayWins=33189; ! plays=1E6; switchWins=667167; stayWins=332833;</lang>
Icon and Unicon
<lang Icon>procedure main(arglist)
rounds := integer(arglist[1]) | 10000 doors := '123' strategy1 := strategy2 := 0
every 1 to rounds do {
goats := doors -- ( car := ?doors ) guess1 := ?doors show := goats -- guess1 if guess1 == car then strategy1 +:= 1 else strategy2 +:= 1 }
write("Monty Hall simulation for ", rounds, " rounds.") write("Strategy 1 'Staying' won ", real(strategy1) / rounds ) write("Strategy 2 'Switching' won ", real(strategy2) / rounds )
end</lang>
Sample Output:
Monty Hall simulation for 10000 rounds. Strategy 1 'Staying' won 0.3266 Strategy 2 'Switching' won 0.6734
Io
<lang Io>keepWins := 0 switchWins := 0 doors := 3 times := 100000 pickDoor := method(excludeA, excludeB,
door := excludeA while(door == excludeA or door == excludeB, door = (Random value() * doors) floor ) door
) times repeat(
playerChoice := pickDoor() carDoor := pickDoor() shownDoor := pickDoor(carDoor, playerChoice) switchDoor := pickDoor(playerChoice, shownDoor) (playerChoice == carDoor) ifTrue(keepWins = keepWins + 1) (switchDoor == carDoor) ifTrue(switchWins = switchWins + 1)
) ("Switching to the other door won #{switchWins} times.\n"\
.. "Keeping the same door won #{keepWins} times.\n"\
.. "Game played #{times} times with #{doors} doors.") interpolate println
</lang>
Sample output:
Switching to the other door won 66935 times. Keeping the same door won 33065 times. Game played 100000 times with 3 doors.
J
The core of this simulation is picking a random item from a set
<lang j>pick=: {~ ?@#</lang>
And, of course, we will be picking one door from three doors
<lang j>DOORS=:1 2 3</lang>
But note that the simulation code should work just as well with more doors.
Anyways the scenario where the contestant's switch or stay strategy makes a difference is where Monty has picked from the doors which are neither the user's door nor the car's door.
<lang j>scenario=: ((pick@-.,])pick,pick) bind DOORS</lang>
(Here, I have decided that the result will be a list of three door numbers. The first number in that list is the number Monty picks, the second number represents the door the user picked, and the third number represents the door where the car is hidden.)
Once we have our simulation test results for the scenario, we need to test if staying would win. In other words we need to test if the user's first choice matches where the car was hidden:
<lang j>stayWin=: =/@}.</lang>
In other words: drop the first element from the list representing our test results -- this leaves us with the user's choice and the door where the car was hidden -- and then insert the verb = between those two values.
We also need to test if switching would win. In other words, we need to test if the user would pick the car from the doors other than the one Monty picked and the one the user originally picked:
<lang j>switchWin=: pick@(DOORS -. }:) = {:</lang>
In other words, start with our list of all doors and then remove the door the monty picked and the door the user picked, and then pick one of the remaining doors at random (the pick at random part is only significant if there were originally more than 3 doors) and see if that matches the door where the car is.
Finally, we need to run the simulation a thousand times and count how many times each strategy wins:
<lang j> +/ (stayWin,switchWin)@scenario"0 i.1000 320 680</lang>
Or, we could bundle this all up as a defined word. Here, the (optional) left argument "names" the doors and the right argument says how many simulations to run:
<lang j>simulate=:3 :0
1 2 3 simulate y
pick=. {~ ?@#
scenario=. ((pick@-.,])pick,pick) bind x
stayWin=. =/@}.
switchWin=. pick@(x -. }:) = {:
r=.(stayWin,switchWin)@scenario"0 i.y
labels=. ];.2 'limit stay switch '
smoutput labels,.":"0 y,+/r
)</lang>
Example use:
<lang j> simulate 1000 limit 1000 stay 304 switch 696 </lang>
Or, with more doors (and assuming this does not require new rules about how Monty behaves or how the player behaves):
<lang j> 1 2 3 4 simulate 1000 limit 1000 stay 233 switch 388 </lang>
Java
<lang java>import java.util.Random; public class Monty{ public static void main(String[] args){ int switchWins = 0; int stayWins = 0; Random gen = new Random(); for(int plays = 0;plays < 32768;plays++ ){ int[] doors = {0,0,0};//0 is a goat, 1 is a car doors[gen.nextInt(3)] = 1;//put a winner in a random door int choice = gen.nextInt(3); //pick a door, any door int shown; //the shown door do{ shown = gen.nextInt(3); //don't show the winner or the choice }while(doors[shown] == 1 || shown == choice);
stayWins += doors[choice];//if you won by staying, count it
//the switched (last remaining) door is (3 - choice - shown), because 0+1+2=3 switchWins += doors[3 - choice - shown]; } System.out.println("Switching wins " + switchWins + " times."); System.out.println("Staying wins " + stayWins + " times."); } }</lang> Output:
Switching wins 21924 times. Staying wins 10844 times.
JavaScript
Extensive Solution
This solution can test with n doors, the difference in probability for switching is shown to diminish as the number of doors increases.
<lang javascript> function montyhall(tests, doors) { 'use strict'; tests = tests ? tests : 1000; doors = doors ? doors : 3; var prizeDoor, chosenDoor, shownDoor, switchDoor, chosenWins = 0, switchWins = 0;
// randomly pick a door excluding input doors function pick(excludeA, excludeB) { var door; do { door = Math.floor(Math.random() * doors); } while (door === excludeA || door === excludeB); return door; }
// run tests for (var i = 0; i < tests; i ++) {
// pick set of doors prizeDoor = pick(); chosenDoor = pick(); shownDoor = pick(prizeDoor, chosenDoor); switchDoor = pick(chosenDoor, shownDoor);
// test set for both choices if (chosenDoor === prizeDoor) { chosenWins ++; } else if (switchDoor === prizeDoor) { switchWins ++; } }
// results return { stayWins: chosenWins + ' ' + (100 * chosenWins / tests) + '%', switchWins: switchWins + ' ' + (100 * switchWins / tests) + '%' }; } </lang>
- Output:
<lang javascript> montyhall(1000, 3) Object {stayWins: "349 34.9%", switchWins: "651 65.1%"} montyhall(1000, 4) Object {stayWins: "253 25.3%", switchWins: "384 38.4%"} montyhall(1000, 5) Object {stayWins: "202 20.2%", switchWins: "265 26.5%"} </lang>
Slight modification of the script above for modularity inside of HTML. <lang javascript><html>
<body>
<input id="userInputMH" value="1000"> <input id="userInputDoor" value="3">
<button onclick="montyhall()">Calculate</button>
</body>
</html>
<script>
function montyhall() {
var tests = document.getElementById("userInputMH").value;
var doors = document.getElementById("userInputDoor").value;
var prizeDoor, chosenDoor, shownDoor, switchDoor, chosenWins = 0,switchWins = 0;
function pick(excludeA, excludeB) {
var door;
do {
door = Math.floor(Math.random() * doors);
} while (door === excludeA || door === excludeB);
return door;
}
for (var i = 0; i < tests; i++) {
prizeDoor = pick();
chosenDoor = pick();
shownDoor = pick(prizeDoor, chosenDoor);
switchDoor = pick(chosenDoor, shownDoor);
if (chosenDoor === prizeDoor) {
chosenWins++;
} else if (switchDoor === prizeDoor) {
switchWins++;
}
}
document.getElementById("firstPickWins").innerHTML = 'First Door Wins: ' + chosenWins + ' | ' + (100 * chosenWins / tests) + '%';
document.getElementById("switchPickWins").innerHTML = 'Switched Door Wins: ' + switchWins + ' | ' + (100 * switchWins / tests) + '%';
}
</script></lang> Output: <lang javascript>(1000, 3) First Door Wins: 346 | 34.6% Switching Door Wins: 654 | 65.4%</lang>
Basic Solution
<lang javascript> var totalGames = 10000,
selectDoor = function () {
return Math.floor(Math.random() * 3); // Choose a number from 0, 1 and 2.
},
games = (function () {
var i = 0, games = [];
for (; i < totalGames; ++i) { games.push(selectDoor()); // Pick a door which will hide the prize. }
return games;
}()),
play = function (switchDoor) {
var i = 0, j = games.length, winningDoor, randomGuess, totalTimesWon = 0;
for (; i < j; ++i) { winningDoor = games[i]; randomGuess = selectDoor(); if ((randomGuess === winningDoor && !switchDoor) || (randomGuess !== winningDoor && switchDoor)) { /* * If I initially guessed the winning door and didn't switch, * or if I initially guessed a losing door but then switched, * I've won. * * I lose when I initially guess the winning door and then switch,
* or initially guess a losing door and don't switch.
*/
totalTimesWon++; } } return totalTimesWon;
};
/*
* Start the simulation */
console.log("Playing " + totalGames + " games"); console.log("Wins when not switching door", play(false)); console.log("Wins when switching door", play(true)); </lang>
- Output:
<lang javascript> Playing 10000 games Wins when not switching door 3326 Wins when switching door 6630 </lang>
jq
Works with gojq, the Go implementation of jq
Neither the C nor the Go implementations of jq currently provides a PRN generator, so this entry uses /dev/urandom as an external source of entropy as follows:
cat /dev/urandom | tr -cd '0-9' | fold -w 1 | jq -nrf monty-hall.jq
where monty-hall.jq contains one of the following jq programs.
Basic solution
This solution is based on the observation:
<lang jq>def rand:
input as $r | if $r < . then $r else rand end;
def logical_montyHall:
. as $games
| {switchWins: 0, stayWins: 0}
| reduce range (0; $games) as $_ (.;
(3|rand) as $car # put car in a random door
| (3|rand) as $choice # choose a door at random
| if $choice == $car then .stayWins += 1
else .switchWins += 1 end )
| "Simulating \($games) games:",
"Staying wins \(.stayWins) times",
"Switching wins \(.switchWins) times\n" ;
1e3, 1e6 | logical_montyHall</lang>
Simulation
<lang jq>def rand:
input as $r | if $r < . then $r else rand end;
def montyHall:
. as $games
| [range(0;3) | 0 ] as $doors0
| {switchWins: 0, stayWins: 0}
| reduce range (0; $games) as $_ (.;
($doors0 | .[3|rand] = 1) as $doors # put car in a random door
| (3|rand) as $choice # choose a door at random
| .stop = false
| until (.stop;
.shown = (3|rand) # the shown door
| if ($doors[.shown] != 1 and .shown != $choice)
then .stop=true else . end)
| .stayWins += $doors[$choice]
| .switchWins += $doors[3 - $choice - .shown]
)
| "Simulating \($games) games:",
"Staying wins \(.stayWins) times",
"Switching wins \(.switchWins) times\n" ;
1e3, 1e6 | montyHall</lang>
- Output:
Simulating 1000 games: Staying wins 325 times Switching wins 675 times Simulating 1000000 games: Staying wins 333253 times Switching wins 666747 times
Julia
To make things interesting, I decided to generalize the problem to ncur doors and ncar cars. To allow the MC to always show a goat behind a door after the contestant chooses, .
I was was of two minds on the type of simulation to provide, so I wrote two different simulators. The literal simulator mimics the mechanics of the game as literally as possible, shuffling the arrangement of cars behind doors and randomizes all choices. This avoids any feel of cheating but results in rather complex code. The clean simulator implements the game more elegantly but it might look like cheating.
The Literal Simulation Function <lang Julia>using Printf
function play_mh_literal{T<:Integer}(ncur::T=3, ncar::T=1)
ncar < ncur || throw(DomainError())
curtains = shuffle(collect(1:ncur))
cars = curtains[1:ncar]
goats = curtains[(ncar+1):end]
pick = rand(1:ncur)
isstickwin = pick in cars
deleteat!(curtains, findin(curtains, pick))
if !isstickwin
deleteat!(goats, findin(goats, pick))
end
if length(goats) > 0 # reveal goat
deleteat!(curtains, findin(curtains, shuffle(goats)[1]))
else # no goats, so reveal car
deleteat!(curtains, rand(1:(ncur-1)))
end
pick = shuffle(curtains)[1]
isswitchwin = pick in cars
return (isstickwin, isswitchwin)
end </lang>
The Clean Simulation Function <lang Julia> function play_mh_clean{T<:Integer}(ncur::T=3, ncar::T=1)
ncar < ncur || throw(DomainError())
pick = rand(1:ncur)
isstickwin = pick <= ncar
pick = rand(1:(ncur-2))
if isstickwin # remove initially picked car from consideration
pick += 1
end
isswitchwin = pick <= ncar
return (isstickwin, isswitchwin)
end </lang>
Supporting Functions <lang Julia> function mh_results{T<:Integer}(ncur::T, ncar::T,
nruns::T, play_mh::Function)
stickwins = 0
switchwins = 0
for i in 1:nruns
(isstickwin, isswitchwin) = play_mh(ncur, ncar)
if isstickwin
stickwins += 1
end
if isswitchwin
switchwins += 1
end
end
return (stickwins/nruns, switchwins/nruns)
end
function mh_analytic{T<:Integer}(ncur::T, ncar::T)
stickodds = ncar/ncur switchodds = (ncar - stickodds)/(ncur-2) return (stickodds, switchodds)
end
function show_odds{T<:Real}(a::T, b::T)
@sprintf " %.1f %.1f %.2f" 100.0*a 100*b 1.0*b/a
end
function show_simulation{T<:Integer}(ncur::T, ncar::T, nruns::T)
println()
print("Simulating a ", ncur, " door, ", ncar, " car ")
println("Monty Hall problem with ", nruns, " runs.\n")
println(" Solution Stick Switch Improvement")
(a, b) = mh_results(ncur, ncar, nruns, play_mh_literal)
println(@sprintf("%10s: ", "Literal"), show_odds(a, b))
(a, b) = mh_results(ncur, ncar, nruns, play_mh_clean)
println(@sprintf("%10s: ", "Clean"), show_odds(a, b))
(a, b) = mh_analytic(ncur, ncar)
println(@sprintf("%10s: ", "Analytic"), show_odds(a, b))
println()
return nothing
end </lang>
Main <lang Julia> for i in 3:5, j in 1:(i-2)
show_simulation(i, j, 10^5)
end </lang>
This code shows, for a variety of configurations, the results for 3 solutions: literal simulation, clean simulation, analytic. Stick is the percentage of times that the player wins a car by sticking to an initial choice. Switch is the winning percentage the comes with switching one's selection following the goat reveal. Improvement is the ratio of switch to stick.
- Output:
Simulating a 3 door, 1 car Monty Hall problem with 100000 runs.
Solution Stick Switch Improvement
Literal: 33.2 66.8 2.02
Clean: 33.4 66.6 2.00
Analytic: 33.3 66.7 2.00
Simulating a 4 door, 1 car Monty Hall problem with 100000 runs.
Solution Stick Switch Improvement
Literal: 25.1 37.5 1.49
Clean: 24.7 37.6 1.52
Analytic: 25.0 37.5 1.50
Simulating a 4 door, 2 car Monty Hall problem with 100000 runs.
Solution Stick Switch Improvement
Literal: 49.7 75.3 1.51
Clean: 49.9 74.9 1.50
Analytic: 50.0 75.0 1.50
Simulating a 5 door, 1 car Monty Hall problem with 100000 runs.
Solution Stick Switch Improvement
Literal: 20.2 26.5 1.31
Clean: 20.0 26.8 1.34
Analytic: 20.0 26.7 1.33
Simulating a 5 door, 2 car Monty Hall problem with 100000 runs.
Solution Stick Switch Improvement
Literal: 40.0 53.5 1.34
Clean: 40.4 53.4 1.32
Analytic: 40.0 53.3 1.33
Simulating a 5 door, 3 car Monty Hall problem with 100000 runs.
Solution Stick Switch Improvement
Literal: 60.3 79.9 1.33
Clean: 59.9 80.1 1.34
Analytic: 60.0 80.0 1.33
Literal versus Clean
The clean simulation runs significantly faster and uses less memory.
julia> @time mh_results(3, 1, 10^5, play_mh_literal) elapsed time: 0.346965522 seconds (183790752 bytes allocated, 27.56% gc time) (0.33216,0.66784) julia> @time mh_results(3, 1, 10^5, play_mh_clean) elapsed time: 0.046481738 seconds (9600160 bytes allocated) (0.33241,0.66759)
Kotlin
<lang scala>// version 1.1.2
import java.util.Random
fun montyHall(games: Int) {
var switchWins = 0
var stayWins = 0
val rnd = Random()
(1..games).forEach {
val doors = IntArray(3) // all zero (goats) by default
doors[rnd.nextInt(3)] = 1 // put car in a random door
val choice = rnd.nextInt(3) // choose a door at random
var shown: Int
do {
shown = rnd.nextInt(3) // the shown door
}
while (doors[shown] == 1 || shown == choice)
stayWins += doors[choice]
switchWins += doors[3 - choice - shown]
}
println("Simulating $games games:")
println("Staying wins $stayWins times")
println("Switching wins $switchWins times")
}
fun main(args: Array<String>) {
montyHall(1_000_000)
}</lang> Sample output:
- Output:
Simulating 1000000 games: Staying wins 333670 times Switching wins 666330 times
Liberty BASIC
<lang lb> 'adapted from BASIC solution DIM doors(3) '0 is a goat, 1 is a car
total = 10000 'set desired number of iterations switchWins = 0 stayWins = 0
FOR plays = 1 TO total
winner = INT(RND(1) * 3) + 1
doors(winner) = 1'put a winner in a random door
choice = INT(RND(1) * 3) + 1'pick a door, any door
DO
shown = INT(RND(1) * 3) + 1
'don't show the winner or the choice
LOOP WHILE doors(shown) = 1 OR shown = choice
if doors(choice) = 1 then
stayWins = stayWins + 1 'if you won by staying, count it
else
switchWins = switchWins + 1'could have switched to win
end if
doors(winner) = 0 'clear the doors for the next test
NEXT PRINT "Result for ";total;" games." PRINT "Switching wins "; switchWins; " times." PRINT "Staying wins "; stayWins; " times." </lang> Output:
Result for 10000 games. Switching wins 6634 times. Staying wins 3366 times.
Lua
<lang lua>function playgame(player)
local car = math.random(3)
local pchoice = player.choice()
local function neither(a, b) --slow, but it works
local el = math.random(3)
return (el ~= a and el ~= b) and el or neither(a, b)
end
local el = neither(car, pchoice)
if(player.switch) then pchoice = neither(pchoice, el) end
player.wins = player.wins + (pchoice == car and 1 or 0)
end for _, v in ipairs{true, false} do
player = {choice = function() return math.random(3) end,
wins = 0, switch = v}
for i = 1, 20000 do playgame(player) end
print(player.wins)
end</lang>
Lua/Torch
<lang lua>function montyHall(n)
local car = torch.LongTensor(n):random(3) -- door with car local choice = torch.LongTensor(n):random(3) -- player's choice local opens = torch.LongTensor(n):random(2):csub(1):mul(2):csub(1) -- -1 or +1 local iscarchoice = choice:eq(car) local nocarchoice = 1-iscarchoice opens[iscarchoice] = (((opens + choice - 1) % 3):abs() + 1)[iscarchoice] opens[nocarchoice] = (6 - car - choice)[nocarchoice] local change = torch.LongTensor(n):bernoulli() -- 0: stay, 1: change local win = iscarchoice:long():cmul(1-change) + nocarchoice:long():cmul(change) return car, choice, opens, change, win
end
function montyStats(n)
local car, pchoice, opens, change, win = montyHall(n) local change_and_win = change [ win:byte()]:sum()/ change :sum()*100 local no_change_and_win = (1-change)[ win:byte()]:sum()/(1-change):sum()*100 local change_and_win_not = change [1-win:byte()]:sum()/ change :sum()*100 local no_change_and_win_not = (1-change)[1-win:byte()]:sum()/(1-change):sum()*100
print(string.format(" %9s %9s" , "no change", "change" ))
print(string.format("win %8.4f%% %8.4f%%", no_change_and_win , change_and_win ))
print(string.format("win not %8.4f%% %8.4f%%", no_change_and_win_not, change_and_win_not))
end
montyStats(1e7)</lang>
Output for 10 million samples:
no change change win 33.3008% 66.6487% win not 66.6992% 33.3513%
M2000 Interpreter
<lang M2000 Interpreter> Module CheckIt {
Enum Strat {Stay, Random, Switch}
total=10000
Print $("0.00")
player_win_stay=0
player_win_switch=0
player_win_random=0
For i=1 to total {
Dim doors(1 to 3)=False
doors(Random(1,3))=True
guess=Random(1,3)
Inventory other
for k=1 to 3 {
If k <> guess Then Append other, k
}
If doors(guess) Then {
Mont_Hall_show=other(Random(0,1)!)
} Else {
If doors(other(0!)) Then {
Mont_Hall_show=other(1!)
} Else Mont_Hall_show=other(0!)
Delete Other, Mont_Hall_show
}
Strategy=Each(Strat)
While Strategy {
Select Case Eval(strategy)
Case Random
{
If Random(1,2)=1 Then {
If doors(guess) Then player_win_Random++
} else If doors(other(0!)) Then player_win_Random++
}
Case Switch
If doors(other(0!)) Then player_win_switch++
Else
If doors(guess) Then player_win_stay++
End Select
}
}
Print "Stay: ";player_win_stay/total*100;"%"
Print "Random: ";player_win_Random/total*100;"%"
Print "Switch: ";player_win_switch/total*100;"%"
} CheckIt </lang>
- Output:
Stay: 33.39% Random: 51.00% Switch: 66.61%
Mathematica/Wolfram Language
<lang Mathematica> montyHall[nGames_] :=
Module[{r, winningDoors, firstChoices, nStayWins, nSwitchWins, s},
r := RandomInteger[{1, 3}, nGames];
winningDoors = r;
firstChoices = r;
nStayWins = Count[Transpose[{winningDoors, firstChoices}], {d_, d_}];
nSwitchWins = nGames - nStayWins;
Grid[{{"Strategy", "Wins", "Win %"}, {"Stay", Row[{nStayWins, "/", nGames}], s=N[100 nStayWins/nGames]},
{"Switch", Row[{nSwitchWins, "/", nGames}], 100 - s}}, Frame -> All]]</lang>
- Usage
<lang Mathematica>montyHall[100000]</lang>
MATLAB
<lang MATLAB>function montyHall(numDoors,numSimulations)
assert(numDoors > 2);
function num = randInt(n)
num = floor( n*rand()+1 );
end
%The first column will tallie wins, the second losses switchedDoors = [0 0]; stayed = [0 0];
for i = (1:numSimulations)
availableDoors = (1:numDoors); %Preallocate the available doors
winningDoor = randInt(numDoors); %Define the winning door
playersOriginalChoice = randInt(numDoors); %The player picks his initial choice
availableDoors(playersOriginalChoice) = []; %Remove the players choice from the available doors
%Pick the door to open from the available doors
openDoor = availableDoors(randperm(numel(availableDoors))); %Sort the available doors randomly
openDoor(openDoor == winningDoor) = []; %Make sure Monty doesn't open the winning door
openDoor = openDoor(randInt(numel(openDoor))); %Choose a random door to open
availableDoors(availableDoors==openDoor) = []; %Remove the open door from the available doors
availableDoors(end+1) = playersOriginalChoice; %Put the player's original choice back into the pool of available doors
availableDoors = sort(availableDoors);
playersNewChoice = availableDoors(randInt(numel(availableDoors))); %Pick one of the available doors
if playersNewChoice == playersOriginalChoice
switch playersNewChoice == winningDoor
case true
stayed(1) = stayed(1) + 1;
case false
stayed(2) = stayed(2) + 1;
otherwise
error 'ERROR'
end
else
switch playersNewChoice == winningDoor
case true
switchedDoors(1) = switchedDoors(1) + 1;
case false
switchedDoors(2) = switchedDoors(2) + 1;
otherwise
error 'ERROR'
end
end
end
disp(sprintf('Switch win percentage: %f%%\nStay win percentage: %f%%\n', [switchedDoors(1)/sum(switchedDoors),stayed(1)/sum(stayed)] * 100));
end</lang>
Output: <lang MATLAB>>> montyHall(3,100000) Switch win percentage: 66.705972% Stay win percentage: 33.420062%</lang>
MAXScript
<lang maxscript>fn montyHall choice switch = (
doors = #(false, false, false) doors[random 1 3] = true chosen = doors[choice] if switch then chosen = not chosen chosen
)
fn iterate iterations switched = (
wins = 0
for i in 1 to iterations do
(
if (montyHall (random 1 3) switched) then
(
wins += 1
)
)
wins * 100 / iterations as float
)
iterations = 10000 format ("Stay strategy:%\%\n") (iterate iterations false) format ("Switch strategy:%\%\n") (iterate iterations true)</lang> Output: <lang maxscript>Stay strategy:33.77% Switch strategy:66.84%</lang>
NetRexx
<lang netrexx>/* NetRexx ************************************************************
- 30.08.2013 Walter Pachl translated from Java/REXX/PL/I
- /
options replace format comments java crossref savelog symbols nobinary
doors = create_doors switchWins = 0 stayWins = 0 shown=0 Loop plays=1 To 1000000
doors=0 r=r3() doors[r]=1 choice = r3() loop Until shown<>choice & doors[shown]=0 shown = r3() End If doors[choice]=1 Then stayWins=stayWins+1 Else switchWins=switchWins+1 End
Say "Switching wins " switchWins " times." Say "Staying wins " stayWins " times."
method create_doors static returns Rexx
doors = doors[0] = 0 doors[1] = 0 doors[2] = 0 return doors
method r3 static
rand=random() return rand.nextInt(3) + 1</lang>
Output
Switching wins 667335 times. Staying wins 332665 times.
Nim
<lang nim>import random
randomize()
proc shuffle[T](x: var seq[T]) =
for i in countdown(x.high, 0): let j = rand(i) swap(x[i], x[j])
- 1 represents a car
- 0 represent a goat
var
stay = 0 # amount won if stay in the same position switch = 0 # amount won if you switch
for i in 1..1000:
var lst = @[1,0,0] # one car and two goats shuffle(lst) # shuffles the list randomly let ran = rand(2 ) # gets a random number for the random guess let user = lst[ran] # storing the random guess del lst, ran # deleting the random guess
var huh = 0
for i in lst: # getting a value 0 and deleting it
if i == 0:
del lst, huh # deletes a goat when it finds it
break
inc huh
if user == 1: # if the original choice is 1 then stay adds 1 inc stay
if lst[0] == 1: # if the switched value is 1 then switch adds 1 inc switch
echo "Stay = ",stay echo "Switch = ",switch</lang> Output:
Stay = 337 Switch = 663
OCaml
<lang ocaml>let trials = 10000
type door = Car | Goat
let play switch =
let n = Random.int 3 in
let d1 = [|Car; Goat; Goat|].(n) in
if not switch then d1
else match d1 with
Car -> Goat
| Goat -> Car
let cars n switch =
let total = ref 0 in
for i = 1 to n do
let prize = play switch in
if prize = Car then
incr total
done;
!total
let () =
let switch = cars trials true
and stay = cars trials false in
let msg strat n =
Printf.printf "The %s strategy succeeds %f%% of the time.\n"
strat (100. *. (float n /. float trials)) in
msg "switch" switch;
msg "stay" stay</lang>
PARI/GP
<lang parigp>test(trials)={
my(stay=0,change=0);
for(i=1,trials,
my(prize=random(3),initial=random(3),opened);
while((opened=random(3))==prize | opened==initial,);
if(prize == initial, stay++, change++)
);
print("Wins when staying: "stay);
print("Wins when changing: "change);
[stay, change]
};
test(1e4)</lang>
Output:
Wins when staying: 3433 Wins when changing: 6567 %1 = [3433, 6567]
Pascal
<lang pascal>program MontyHall;
uses
sysutils;
const
NumGames = 1000;
{Randomly pick a door(a number between 0 and 2}
function PickDoor(): Integer;
begin
Exit(Trunc(Random * 3));
end;
var
i: Integer; PrizeDoor: Integer; ChosenDoor: Integer; WinsChangingDoors: Integer = 0; WinsNotChangingDoors: Integer = 0;
begin
Randomize; for i := 0 to NumGames - 1 do begin //randomly picks the prize door PrizeDoor := PickDoor; //randomly chooses a door ChosenDoor := PickDoor;
//if the strategy is not changing doors the only way to win is if the chosen
//door is the one with the prize
if ChosenDoor = PrizeDoor then
Inc(WinsNotChangingDoors);
//if the strategy is changing doors the only way to win is if we choose one
//of the two doors that hasn't the prize, because when we change we change to the prize door.
//The opened door doesn't have a prize
if ChosenDoor <> PrizeDoor then
Inc(WinsChangingDoors);
end;
Writeln('Num of games:' + IntToStr(NumGames));
Writeln('Wins not changing doors:' + IntToStr(WinsNotChangingDoors) + ', ' +
FloatToStr((WinsNotChangingDoors / NumGames) * 100) + '% of total.');
Writeln('Wins changing doors:' + IntToStr(WinsChangingDoors) + ', ' +
FloatToStr((WinsChangingDoors / NumGames) * 100) + '% of total.');
end. </lang>
Output:
Num of games:1000 Wins not changing doors:359, 35,9% of total. Wins changing doors:641, 64,1% of total.
Perl
<lang perl>#! /usr/bin/perl use strict; my $trials = 10000;
my $stay = 0; my $switch = 0;
foreach (1 .. $trials) {
my $prize = int(rand 3);
# let monty randomly choose a door where he puts the prize
my $chosen = int(rand 3);
# let us randomly choose a door...
my $show;
do { $show = int(rand 3) } while $show == $chosen || $show == $prize;
# ^ monty opens a door which is not the one with the
# prize, that he knows it is the one the player chosen
$stay++ if $prize == $chosen;
# ^ if player chose the correct door, player wins only if he stays
$switch++ if $prize == 3 - $chosen - $show;
# ^ if player switches, the door he picks is (3 - $chosen - $show),
# because 0+1+2=3, and he picks the only remaining door that is
# neither $chosen nor $show
}
print "Stay win ratio " . (100.0 * $stay/$trials) . "\n"; print "Switch win ratio " . (100.0 * $switch/$trials) . "\n";</lang>
Phix
Modified copy of Euphoria
with javascript_semantics integer swapWins = 0, stayWins = 0, winner, choice, reveal, other atom t0 = time() for game=1 to 1_000_000 do winner = rand(3) choice = rand(3) while 1 do reveal = rand(3) if reveal!=winner and reveal!=choice then exit end if end while stayWins += (choice==winner) other = 6-choice-reveal -- (as 1+2+3=6, and reveal!=choice) swapWins += (other==winner) end for printf(1, "Stay: %,d\nSwap: %,d\nTime: %s\n",{stayWins,swapWins,elapsed(time()-t0)})
- Output:
Stay: 333,292 Swap: 666,708 Time: 0.2s
PHP
<lang php><?php function montyhall($iterations){ $switch_win = 0; $stay_win = 0;
foreach (range(1, $iterations) as $i){ $doors = array(0, 0, 0); $doors[array_rand($doors)] = 1; $choice = array_rand($doors); do { $shown = array_rand($doors); } while($shown == $choice || $doors[$shown] == 1);
$stay_win += $doors[$choice]; $switch_win += $doors[3 - $choice - $shown]; }
$stay_percentages = ($stay_win/$iterations)*100; $switch_percentages = ($switch_win/$iterations)*100;
echo "Iterations: {$iterations} - "; echo "Stayed wins: {$stay_win} ({$stay_percentages}%) - "; echo "Switched wins: {$switch_win} ({$switch_percentages}%)"; }
montyhall(10000);
?></lang> Output:
Iterations: 10000 - Stayed wins: 3331 (33.31%) - Switched wins: 6669 (66.69%)
PicoLisp
<lang PicoLisp>(de montyHall (Keep)
(let (Prize (rand 1 3) Choice (rand 1 3))
(if Keep # Keeping the first choice?
(= Prize Choice) # Yes: Monty's choice doesn't matter
(<> Prize Choice) ) ) ) # Else: Win if your first choice was wrong
(prinl
"Strategy KEEP -> "
(let Cnt 0
(do 10000 (and (montyHall T) (inc 'Cnt)))
(format Cnt 2) )
" %" )
(prinl
"Strategy SWITCH -> "
(let Cnt 0
(do 10000 (and (montyHall NIL) (inc 'Cnt)))
(format Cnt 2) )
" %" )</lang>
Output:
Strategy KEEP -> 33.01 % Strategy SWITCH -> 67.73 %
PL/I
<lang pli>*process source attributes xref;
ziegen: Proc Options(main);
/* REXX ***************************************************************
* 30.08.2013 Walter Pachl derived from Java
**********************************************************************/
Dcl (switchWins,stayWins) Bin Fixed(31) Init(0);
Dcl doors(3) Bin Fixed(31);
Dcl (plays,r,choice) Bin Fixed(31) Init(0);
Dcl c17 Char(17) Init((datetime()));
Dcl p9 Pic'(9)9' def(c17) pos(5);
i=random(p9);
Do plays=1 To 1000000;
doors=0;
r=r3();
doors(r)=1;
choice=r3();
Do Until(shown^=choice & doors(shown)=0);
shown=r3();
End;
If doors(choice)=1 Then
stayWins+=1;
Else
switchWins+=1;
End;
Put Edit("Switching wins ",switchWins," times.")(Skip,a,f(6),a);
Put Edit("Staying wins ",stayWins ," times.")(Skip,a,f(6),a);
r3: Procedure Returns(Bin Fixed(31)); /********************************************************************* * Return a random integer: 1, 2, or 3 *********************************************************************/ Dcl r Bin Float(53); Dcl res Bin Fixed(31); r=random(); res=(r*3)+1; Return(res); End; End;</lang>
Output:
Switching wins 665908 times. Staying wins 334092 times.
PostScript
Use ghostscript or print this to a postscript printer
<lang PostScript>%!PS /Courier % name the desired font 20 selectfont % choose the size in points and establish
% the font as the current one
% init random number generator (%Calendar%) currentdevparams /Second get srand
1000000 % iteration count 0 0 % 0 wins on first selection 0 wins on switch 2 index % get iteration count { rand 3 mod % winning door rand 3 mod % first choice eq { 1 add } { exch 1 add exch } ifelse } repeat
% compute percentages 2 index div 100 mul exch 2 index div 100 mul
% display result
70 600 moveto
(Switching the door: ) show
80 string cvs show (%) show
70 700 moveto
(Keeping the same: ) show
80 string cvs show (%) show
showpage % print all on the page</lang>
Sample output:
Keeping the same: 33.4163% Switching the door: 66.5837%
PowerShell
<lang Powershell>#Declaring variables $intIterations = 10000 $intKept = 0 $intSwitched = 0
- Creating a function
Function Play-MontyHall()
{
#Using a .NET object for randomization
$objRandom = New-Object -TypeName System.Random
#Generating the winning door number
$intWin = $objRandom.Next(1,4)
#Generating the chosen door
$intChoice = $objRandom.Next(1,4)
#Generating the excluded number
#Because there is no method to exclude a number from a range,
#I let it re-generate in case it equals the winning number or
#in case it equals the chosen door.
$intLose = $objRandom.Next(1,4)
While (($intLose -EQ $intWin) -OR ($intLose -EQ $intChoice))
{$intLose = $objRandom.Next(1,4)}
#Generating the 'other' door
#Same logic applies as for the chosen door: it cannot be equal
#to the winning door nor to the chosen door.
$intSwitch = $objRandom.Next(1,4)
While (($intSwitch -EQ $intLose) -OR ($intSwitch -EQ $intChoice))
{$intSwitch = $objRandom.Next(1,4)}
#Simple counters per win for both categories
#Because a child scope cannot change variables in the parent
#scope, the scope of the counters is expanded script-wide.
If ($intChoice -EQ $intWin)
{$script:intKept++}
If ($intSwitch -EQ $intWin)
{$script:intSwitched++}
}
- Looping the Monty Hall function for $intIterations times
While ($intIterationCount -LT $intIterations)
{
Play-MontyHall
$intIterationCount++
}
- Output
Write-Host "Results through $intIterations iterations:" Write-Host "Keep : $intKept ($($intKept/$intIterations*100)%)" Write-Host "Switch: $intSwitched ($($intSwitched/$intIterations*100)%)" Write-Host ""</lang> Output:
Results through 10000 iterations: Keep : 3336 (33.36%) Switch: 6664 (66.64%)
PureBasic
<lang PureBasic>Structure wins
stay.i redecide.i
EndStructure
- goat = 0
- car = 1
Procedure MontyHall(*results.wins)
Dim Doors(2) Doors(Random(2)) = #car
player = Random(2)
Select Doors(player)
Case #car
*results\redecide + #goat
*results\stay + #car
Case #goat
*results\redecide + #car
*results\stay + #goat
EndSelect
EndProcedure
OpenConsole()
- Tries = 1000000
Define results.wins
For i = 1 To #Tries
MontyHall(@results)
Next
PrintN("Trial runs for each option: " + Str(#Tries)) PrintN("Wins when redeciding: " + Str(results\redecide) + " (" + StrD(results\redecide / #Tries * 100, 2) + "% chance)") PrintN("Wins when sticking: " + Str(results\stay) + " (" + StrD(results\stay / #Tries * 100, 2) + "% chance)") Input()</lang>
Output:
Trial runs for each option: 1000000 Wins when redeciding: 666459 (66.65% chance) Wins when sticking: 333541 (33.35% chance)
Python
<lang python> I could understand the explanation of the Monty Hall problem but needed some more evidence
References:
http://www.bbc.co.uk/dna/h2g2/A1054306 http://en.wikipedia.org/wiki/Monty_Hall_problem especially: http://en.wikipedia.org/wiki/Monty_Hall_problem#Increasing_the_number_of_doors
from random import randrange
doors, iterations = 3,100000 # could try 100,1000
def monty_hall(choice, switch=False, doorCount=doors):
# Set up doors door = [False]*doorCount # One door with prize door[randrange(doorCount)] = True
chosen = door[choice]
unpicked = door del unpicked[choice]
# Out of those unpicked, the alternative is either: # the prize door, or # an empty door if the initial choice is actually the prize. alternative = True in unpicked
if switch: return alternative else: return chosen
print "\nMonty Hall problem simulation:" print doors, "doors,", iterations, "iterations.\n"
print "Not switching allows you to win", print sum(monty_hall(randrange(3), switch=False)
for x in range(iterations)),
print "out of", iterations, "times." print "Switching allows you to win", print sum(monty_hall(randrange(3), switch=True)
for x in range(iterations)),
print "out of", iterations, "times.\n"</lang> Sample output:
Monty Hall problem simulation: 3 doors, 100000 iterations. Not switching allows you to win 33337 out of 100000 times. Switching allows you to win 66529 out of 100000 times.
Python 3 version:
Another (simpler in my opinion), way to do this is below, also in python 3: <lang python>import random
#1 represents a car #0 represent a goat
stay = 0 #amount won if stay in the same position switch = 0 # amount won if you switch
for i in range(1000):
lst = [1,0,0] # one car and two goats random.shuffle(lst) # shuffles the list randomly ran = random.randrange(3) # gets a random number for the random guess
user = lst[ran] #storing the random guess
del(lst[ran]) # deleting the random guess
huh = 0
for i in lst: # getting a value 0 and deleting it
if i ==0:
del(lst[huh]) # deletes a goat when it finds it
break
huh+=1
if user ==1: # if the original choice is 1 then stay adds 1
stay+=1
if lst[0] == 1: # if the switched value is 1 then switch adds 1
switch+=1
print("Stay =",stay) print("Switch = ",switch)
- Done by Sam Witton 09/04/2014</lang>
Quackery
<lang Quackery> [ $ "bigrat.qky" loadfile ] now!
[ 0 ( number of cars when not changing choice )
0 ( number of cars when changing choice )
rot times
[ 3 random ( door with goat )
3 random ( contestant's choice )
= ( If the two numbers are equal then the contestant
wins a car if they change their mind, and they win
a goat if they don't change their mind. The wins
are reversed if the numbers are not equal. )
if dip 1+ ] ( increment the relevant count )
say "Strategy A is that the contestant changes their mind." cr
say "Strategy B is that the contestant does not their mind." cr
say "Approximate ratio of car wins with strategy A over strategy B: "
swap 100 round
vulgar$ echo$ cr ] is trials ( n --> )</lang>
- Output:
Running the simulation in the Quackery shell.
/O> 1000 trials ... Strategy A is that the contestant changes their mind. Strategy B is that the contestant does not their mind. Approximate ratio of car wins with strategy A over strategy B: 15/8 Stack empty. /O> 1000000 trials ... Strategy A is that the contestant changes their mind. Strategy B is that the contestant does not their mind. Approximate ratio of car wins with strategy A over strategy B: 2/1 Stack empty.
From this we can conclude that strategy B is preferable, as goats are the GOAT!
R
<lang rsplus># Since R is a vector based language that penalizes for loops, we will avoid
- for-loops, instead using "apply" statement variants (like "map" in other
- functional languages).
set.seed(19771025) # set the seed to set the same results as this code N <- 10000 # trials true_answers <- sample(1:3, N, replace=TRUE)
- We can assme that the contestant always choose door 1 without any loss of
- generality, by equivalence. That is, we can always relabel the doors
- to make the user-chosen door into door 1.
- Thus, the host opens door '2' unless door 2 has the prize, in which case
- the host opens door 3.
host_opens <- 2 + (true_answers == 2) other_door <- 2 + (true_answers != 2)
- if always switch
summary( other_door == true_answers )
- if we never switch
summary( true_answers == 1)
- if we randomly switch
random_switch <- other_door random_switch[runif(N) >= .5] <- 1 summary(random_switch == true_answers)
- To go with the exact parameters of the Rosetta challenge, complicating matters....
- Note that the player may initially choose any of the three doors (not just Door 1),
- that the host opens a different door revealing a goat (not necessarily Door 3), and
- that he gives the player a second choice between the two remaining unopened doors.
N <- 10000 #trials true_answers <- sample(1:3, N, replace=TRUE) user_choice <- sample(1:3, N, replace=TRUE)
- the host_choice is more complicated
host_chooser <- function(user_prize) {
# this could be cleaner
bad_choices <- unique(user_prize)
# in R, the x[-vector] form implies, choose the indices in x not in vector
choices <- c(1:3)[-bad_choices]
# if the first arg to sample is an int, it treats it as the number of choices
if (length(choices) == 1) { return(choices)}
else { return(sample(choices,1))}
}
host_choice <- apply( X=cbind(true_answers,user_choice), FUN=host_chooser,MARGIN=1) not_door <- function(x){ return( (1:3)[-x]) } # we could also define this
# directly at the FUN argument following
other_door <- apply( X = cbind(user_choice,host_choice), FUN=not_door, MARGIN=1)
- if always switch
summary( other_door == true_answers )
- if we never switch
summary( true_answers == user_choice)
- if we randomly switch
random_switch <- user_choice change <- runif(N) >= .5 random_switch[change] <- other_door[change] summary(random_switch == true_answers)</lang>
Results: > ## if always switch > summary( other_door == true_answers ) Mode FALSE TRUE logical 3298 6702 > ## if we never switch > summary( true_answers == 1) Mode FALSE TRUE logical 6702 3298 > ## if we randomly switch > summary(random_switch == true_answers) Mode FALSE TRUE logical 5028 4972 > ## if always switch > summary( other_door == true_answers ) Mode FALSE TRUE logical 3295 6705 > ## if we never switch > summary( true_answers == user_choice) Mode FALSE TRUE logical 6705 3295 > ## if we randomly switch > summary(random_switch == true_answers) Mode FALSE TRUE logical 4986 5014
# As above, but generalized to K number of doors
K = 4 # number of doors
N = 1e4 # number of simulation trials
chooser <- function(x) { i <- (1:K)[-x]; if (length(i)>1) sample(i,1) else i }
p100 <- function(...) { cat("\nNumber of doors:", K,
"\nSimulation yields % winning probability:",
" (2nd choice after host reveal)\n");
print(c(...) * 100, digits=3) }
prize_door <- sample(1:K, N, replace=TRUE)
first_choice <- sample(1:K, N, replace=TRUE)
host_opens <- apply(cbind(prize_door, first_choice), 1, chooser)
second_choice <- apply(cbind(host_opens, first_choice), 1, chooser)
p100("By first choice" = (Pr.first_win <- mean(first_choice == prize_door)),
"By second choice" = (Pr.second_win <- mean(second_choice == prize_door)),
" Change gain" = Pr.second_win / Pr.first_win - 1)
#-------
#
# Sample output:
Number of doors: 4
Simulation yields % winning probability: (2nd choice after host reveal)
By first choice By second choice Change gain
24.7 36.5 48.0
Racket
<lang Racket>
- lang racket
(define (get-last-door a b) ; assumes a != b
(vector-ref '#(- 2 1
2 - 0
1 0 -)
(+ a (* 3 b))))
(define (run-game strategy)
(define car-door (random 3))
(define first-choice (random 3))
(define revealed-goat
(if (= car-door first-choice)
(let ([r (random 2)]) (if (<= car-door r) (add1 r) r)) ; random
(get-last-door car-door first-choice))) ; reveal goat
(define final-choice (strategy first-choice revealed-goat))
(define win? (eq? final-choice car-door))
;; (printf "car: ~s\nfirst: ~s\nreveal: ~s\nfinal: ~s\n => ~s\n\n"
;; car-door first-choice revealed-goat final-choice
;; (if win? 'win 'lose))
win?)
(define (keep-choice first-choice revealed-goat)
first-choice)
(define (change-choice first-choice revealed-goat)
(get-last-door first-choice revealed-goat))
(define (test-strategy strategy)
(define N 10000000)
(define wins (for/sum ([i (in-range N)]) (if (run-game strategy) 1 0)))
(printf "~a: ~a%\n"
(object-name strategy)
(exact->inexact (/ wins N 1/100))))
(for-each test-strategy (list keep-choice change-choice)) </lang>
Sample Output:
keep-choice: 33.33054% change-choice: 66.67613%
Raku
(formerly Perl 6)
This implementation is parametric over the number of doors. Increasing the number of doors in play makes the superiority of the switch strategy even more obvious.
<lang perl6>enum Prize <Car Goat>; enum Strategy <Stay Switch>;
sub play (Strategy $strategy, Int :$doors = 3) returns Prize {
# Call the door with a car behind it door 0. Number the
# remaining doors starting from 1.
my Prize @doors = flat Car, Goat xx $doors - 1;
# The player chooses a door.
my Prize $initial_pick = @doors.splice(@doors.keys.pick,1)[0];
# Of the n doors remaining, the host chooses n - 1 that have
# goats behind them and opens them, removing them from play.
while @doors > 1 {
@doors.splice($_,1) when Goat given @doors.keys.pick;
} # If the player stays, they get their initial pick. Otherwise, # they get whatever's behind the remaining door. return $strategy === Stay ?? $initial_pick !! @doors[0];
}
constant TRIALS = 10_000;
for 3, 10 -> $doors {
my atomicint @wins[2];
say "With $doors doors: ";
for Stay, 'Staying', Switch, 'Switching' -> $s, $name {
(^TRIALS).race.map: {
@wins[$s]⚛++ if play($s, doors => $doors) == Car;
}
say " $name wins ",
round(100*@wins[$s] / TRIALS),
'% of the time.'
}
}</lang>
- Output:
With 3 doors: Staying wins 34% of the time. Switching wins 66% of the time. With 10 doors: Staying wins 10% of the time. Switching wins 90% of the time.
REXX
version 1
<lang rexx>/* REXX ***************************************************************
- 30.08.2013 Walter Pachl derived from Java
- /
Call time 'R' switchWins = 0; stayWins = 0 Do plays = 1 To 1000000
doors.=0 r=r3() doors.r=1 choice = r3() Do Until shown<>choice & doors.shown=0 shown = r3() End If doors.choice=1 Then stayWins=stayWins+1 Else switchWins=switchWins+1 End
Say "Switching wins " switchWins " times." Say "Staying wins " stayWins " times." Say 'REXX:' time('E') 'seconds' Call time 'R' 'ziegen' Say 'PL/I:' time('E') 'seconds' Say ' ' Call time 'R' 'java ziegen' Say 'NetRexx:' time('E') 'seconds' Exit r3: Return random(2)+1</lang> Output for 1000000 samples:
Switching wins 666442 times. Staying wins 333558 times. REXX: 4.321000 seconds Switching wins 665908 times. Staying wins 334092 times. PL/I: 0.328000 seconds Switching wins 667335 times. Staying wins 332665 times. NetRexx: 2.042000 seconds
From the Rosetta Code:Village Pump/Run times on examples?
As per Michael Mol about showing timings for program execution times:
- Discouraging timing comparisons between different languages.
- Allowing detailed timings, if someone wants to, in the talk pages.
- But generally - like now, leaving them out.
version 2
<lang rexx>/*REXX program simulates any number of trials of the classic TV show Monty Hall problem.*/ parse arg # seed . /*obtain the optional args from the CL.*/ if #== | #=="," then #= 1000000 /*Not specified? Then 1 million trials*/ if datatype(seed, 'W') then call random ,, seed /*Specified? Use as a seed for RANDOM.*/ wins.= 0 /*wins.0 ≡ stay, wins.1 ≡ switching.*/
do #; door. = 0 /*initialize all doors to a value of 0.*/
car= random(1, 3); door.car= 1 /*the TV show hides a car randomly. */
?= random(1, 3); _= door.? /*the contestant picks a (random) door.*/
wins._ = wins._ + 1 /*bump count of type of win strategy.*/
end /*#*/ /* [↑] perform the loop # times. */
/* [↑] door values: 0≡goat 1≡car */
say 'switching wins ' format(wins.0 / # * 100, , 1)"% of the time." say ' staying wins ' format(wins.1 / # * 100, , 1)"% of the time." ; say say 'performed ' # " times with 3 doors." /*stick a fork in it, we're all done. */</lang>
- output when using the default inputs:
switching wins 66.7% of the time. staying wins 33.3% of the time. performed 1000000 times with 3 doors.
Ring
<lang ring> total = 10000 swapper = 0 sticker = 0 revealdoor = 0 for trial = 1 to total
prizedoor = random(3) + 1
guessdoor = random(3) + 1
if prizedoor = guessdoor
revealdoor = random(2) + 1
if prizedoor = 1 revealdoor += 1 ok
if (prizedoor = 2 and revealdoor = 2) revealdoor = 3 ok
else
revealdoor = (prizedoor ^ guessdoor)
ok
stickdoor = guessdoor
swapdoor = (guessdoor ^ revealdoor)
if stickdoor = prizedoor sticker += 1 ok
if swapdoor = prizedoor swapper += 1 ok
next see "after a total of " + total + " trials," + nl see "the 'sticker' won " + sticker + " times (" + floor(sticker/total*100) + "%)" + nl see "the 'swapper' won " + swapper + " times (" + floor(swapper/total*100) + "%)" + nl </lang> Output:
after a total of 10000 trials, the 'sticker' won 2461 times (24%) the 'swapper' won 7539 times (75%)
Ruby
<lang ruby>n = 10_000 #number of times to play
stay = switch = 0 #sum of each strategy's wins
n.times do #play the game n times
#the doors reveal 2 goats and a car doors = [ :goat, :goat, :car ].shuffle #random guess guess = rand(3) #random door shown, but it is neither the guess nor the car begin shown = rand(3) end while shown == guess || doors[shown] == :car if doors[guess] == :car #staying with the initial guess wins if the initial guess is the car stay += 1 else #switching guesses wins if the unshown door is the car switch += 1 end
end
puts "Staying wins %.2f%% of the time." % (100.0 * stay / n) puts "Switching wins %.2f%% of the time." % (100.0 * switch / n)</lang> Sample Output:
Staying wins 33.84% of the time. Switching wins 66.16% of the time.
Run BASIC
<lang runbasic>' adapted from BASIC solution
input "Number of tries;";tries ' gimme the number of iterations FOR plays = 1 TO tries
winner = INT(RND(1) * 3) + 1
doors(winner) = 1 'put a winner in a random door
choice = INT(RND(1) * 3) + 1 'pick a door please
[DO] shown = INT(RND(1) * 3) + 1 ' ------------------------------------------ ' don't show the winner or the choice
if doors(shown) = 1 then goto [DO]
if shown = choice then goto [DO]
if doors(choice) = 1 then
stayWins = stayWins + 1 ' if you won by staying, count it
else
switchWins = switchWins + 1 ' could have switched to win
end if
doors(winner) = 0 'clear the doors for the next test
NEXT PRINT " Result for ";tries;" games." PRINT "Switching wins ";switchWins; " times." PRINT " Staying wins ";stayWins; " times."</lang>
Rust
<lang rust>extern crate rand; use rand::Rng; use rand::seq::SliceRandom;
- [derive(Clone, Copy, PartialEq)]
enum Prize {Goat , Car}
const GAMES: usize = 3_000_000; fn main() {
let mut switch_wins = 0;
let mut rng = rand::thread_rng();
for _ in 0..GAMES {
let mut doors = [Prize::Goat; 3];
*doors.choose_mut(&mut rng).unwrap() = Prize::Car;
// You only lose by switching if you pick the car the first time
if doors.choose(&mut rng).unwrap() != &Prize::Car {
switch_wins += 1;
}
}
println!("I played the game {total} times and won {wins} times ({percent}%).",
total = GAMES,
wins = switch_wins,
percent = switch_wins as f64 / GAMES as f64 * 100.0
);
}</lang>
Scala
<lang scala>import scala.util.Random
object MontyHallSimulation {
def main(args: Array[String]) {
val samples = if (args.size == 1 && (args(0) matches "\\d+")) args(0).toInt else 1000
val doors = Set(0, 1, 2)
var stayStrategyWins = 0
var switchStrategyWins = 0
1 to samples foreach { _ =>
val prizeDoor = Random shuffle doors head;
val choosenDoor = Random shuffle doors head;
val hostDoor = Random shuffle (doors - choosenDoor - prizeDoor) head;
val switchDoor = doors - choosenDoor - hostDoor head;
(choosenDoor, switchDoor) match {
case (`prizeDoor`, _) => stayStrategyWins += 1
case (_, `prizeDoor`) => switchStrategyWins += 1
}
}
def percent(n: Int) = n * 100 / samples
val report = """|%d simulations were ran.
|Staying won %d times (%d %%)
|Switching won %d times (%d %%)""".stripMargin
println(report
format (samples,
stayStrategyWins, percent(stayStrategyWins),
switchStrategyWins, percent(switchStrategyWins)))
}
}</lang>
Sample:
1000 simulations were ran. Staying won 333 times (33 %) Switching won 667 times (66 %)
Scheme
<lang scheme>(define (random-from-list list) (list-ref list (random (length list)))) (define (random-permutation list)
(if (null? list)
'()
(let* ((car (random-from-list list))
(cdr (random-permutation (remove car list))))
(cons car cdr))))
(define (random-configuration) (random-permutation '(goat goat car))) (define (random-door) (random-from-list '(0 1 2)))
(define (trial strategy)
(define (door-with-goat-other-than door strategy)
(cond ((and (not (= 0 door)) (equal? (list-ref strategy 0) 'goat)) 0)
((and (not (= 1 door)) (equal? (list-ref strategy 1) 'goat)) 1)
((and (not (= 2 door)) (equal? (list-ref strategy 2) 'goat)) 2)))
(let* ((configuration (random-configuration))
(players-first-guess (strategy `(would-you-please-pick-a-door?)))
(door-to-show-player (door-with-goat-other-than players-first-guess
configuration))
(players-final-guess (strategy `(there-is-a-goat-at/would-you-like-to-move?
,players-first-guess
,door-to-show-player))))
(if (equal? (list-ref configuration players-final-guess) 'car)
'you-win!
'you-lost)))
(define (stay-strategy message)
(case (car message)
((would-you-please-pick-a-door?) (random-door))
((there-is-a-goat-at/would-you-like-to-move?)
(let ((first-choice (cadr message)))
first-choice))))
(define (switch-strategy message)
(case (car message)
((would-you-please-pick-a-door?) (random-door))
((there-is-a-goat-at/would-you-like-to-move?)
(let ((first-choice (cadr message))
(shown-goat (caddr message)))
(car (remove first-choice (remove shown-goat '(0 1 2))))))))
(define-syntax repeat
(syntax-rules ()
((repeat <n> <body> ...)
(let loop ((i <n>))
(if (zero? i)
'()
(cons ((lambda () <body> ...))
(loop (- i 1))))))))
(define (count element list)
(if (null? list)
0
(if (equal? element (car list))
(+ 1 (count element (cdr list)))
(count element (cdr list)))))
(define (prepare-result strategy results)
`(,strategy won with probability
,(exact->inexact (* 100 (/ (count 'you-win! results) (length results)))) %))
(define (compare-strategies times)
(append (prepare-result 'stay-strategy (repeat times (trial stay-strategy))) '(and) (prepare-result 'switch-strategy (repeat times (trial switch-strategy)))))
- > (compare-strategies 1000000)
- (stay-strategy won with probability 33.3638 %
- and switch-strategy won with probability 66.716 %)</lang>
Scilab
<lang>// How it works: // MontyHall() is a function with argument switch: // it will be called 100000 times with switch=%T, // and another 100000 times with switch=%F
function win=MontyHall(switch) //If switch==%T the player will switch
doors=zeros(1,3) //All goats
car=grand(1,1,'uin',1,3)
a(car)=1 //Place a car somewher
pick=grand(1,1,'uin',1,3) //The player picks...
if pick==car then //If the player picks right...
if switch==%T then //...and switches he will be wrong
win=%F
else //...but if he doesn't, he will be right
win=%T
end
else //If the player picks a goat...
if switch==%T then //...and switches: the other door with the goat shall be
win=%T // opened: the player will switch to the car and win
else //...but if he doesn't, he will remain by his goat
win=%F
end
end
endfunction
wins_switch=0 wins_stay=0 games=100000 for i=1:games
if MontyHall(%T)==%T then
wins_switch=wins_switch+1
end
if MontyHall(%F)==%T then
wins_stay=wins_stay+1
end
end disp("Switching, one wins"+ascii(10)+string(wins_switch)+" games out of "+string(games)) disp("Staying, one wins"+ascii(10)+string(wins_stay)+" games out of "+string(games))</lang>
Output:
Switching, one wins 66649 games out of 100000 Staying, one wins 33403 games out of 100000
Seed7
<lang seed7>$ include "seed7_05.s7i";
const proc: main is func
local
var integer: switchWins is 0;
var integer: stayWins is 0;
var integer: winner is 0;
var integer: choice is 0;
var integer: shown is 0;
var integer: plays is 0;
begin
for plays range 1 to 10000 do
winner := rand(1, 3);
choice := rand(1, 3);
repeat
shown := rand(1, 3)
until shown <> winner and shown <> choice;
stayWins +:= ord(choice = winner);
switchWins +:= ord(6 - choice - shown = winner);
end for;
writeln("Switching wins " <& switchWins <& " times");
writeln("Staying wins " <& stayWins <& " times");
end func;</lang>
Output:
Switching wins 6654 times Staying wins 3346 times
Sidef
<lang ruby>var n = 1000 # number of times to play var switchWins = (var stayWins = 0) # sum of each strategy's wins n.times { # play the game n times
var prize = pick(^3) var chosen = pick(^3)
var show;
do {
show = pick(^3)
} while (show ~~ [chosen, prize])
given(chosen) {
when (prize) { stayWins += 1 }
when ([3 - show - prize]) { switchWins += 1 }
default { die "~ error ~" }
}
} say ("Staying wins %.2f%% of the time." % (100.0 * stayWins / n)) say ("Switching wins %.2f%% of the time." % (100.0 * switchWins / n))</lang>
- Output:
Staying wins 31.20% of the time. Switching wins 68.80% of the time.
SPAD
<lang SPAD> montyHall(n) ==
wd:=[1+random(3) for j in 1..n] fc:=[1+random(3) for j in 1..n] st:=reduce(_+,[1 for j in 1..n | wd.j=fc.j]) p:=(st/n)::DoubleFloat FORMAT(nil,"stay: ~A, switch: ~A",p,1-p)$Lisp
</lang> Domain:Integer
- Output:
(1) -> montyHall(1000)
Compiling function montyHall with type PositiveInteger ->
SExpression
(1) stay: 0.319, switch: 0.681
Type: SExpression
(2) -> montyHall(10000)
(2) stay: 0.3286, switch: 0.6714
Type: SExpression
(3) -> montyHall(100000)
(3) stay: 0.33526, switch: 0.66474
Type: SExpression
Stata
<lang stata>clear set obs 1000000 gen car=runiformint(1,3) gen choice1=runiformint(1,3) gen succ1=car==choice1 gen shown=cond(succ1,runiformint(1,2),6-car-choice1) replace shown=shown+1 if succ1 & (car==1 | car==shown) gen choice2=6-shown-choice1 gen succ2=car==choice2 tabstat succ1 succ2, s(mean)</lang>
Output
stats | succ1 succ2
---------+--------------------
mean | .333632 .666368
------------------------------
Swift
<lang swift>import Foundation
func montyHall(doors: Int = 3, guess: Int, switch: Bool) -> Bool {
guard doors > 2, guess > 0, guess <= doors else { fatalError() }
let winningDoor = Int.random(in: 1...doors)
return winningDoor == guess ? !`switch` : `switch`
}
var switchResults = [Bool]()
for _ in 0..<1_000 {
let guess = Int.random(in: 1...3) let wasRight = montyHall(guess: guess, switch: true)
switchResults.append(wasRight)
}
let switchWins = switchResults.filter({ $0 }).count
print("Switching would've won \((Double(switchWins) / Double(switchResults.count)) * 100)% of games") print("Not switching would've won \(((Double(switchResults.count - switchWins)) / Double(switchResults.count)) * 100)% of games")</lang>
- Output:
Switching would've won 66.8% of games Not switching would've won 33.2% of games
Tcl
A simple way of dealing with this one, based on knowledge of the underlying probabilistic system, is to use code like this: <lang tcl>set stay 0; set change 0; set total 10000 for {set i 0} {$i<$total} {incr i} {
if {int(rand()*3) == int(rand()*3)} {
incr stay
} else {
incr change
}
} puts "Estimate: $stay/$total wins for staying strategy" puts "Estimate: $change/$total wins for changing strategy"</lang> But that's not really the point of this challenge; it should add the concealing factors too so that we're simulating not just the solution to the game, but also the game itself. (Note that we are using Tcl's lists here to simulate sets.)
We include a third strategy that is proposed by some people (who haven't thought much about it) for this game: just picking at random between all the doors offered by Monty the second time round. <lang tcl>package require Tcl 8.5
- Utility: pick a random item from a list
proc pick list {
lindex $list [expr {int(rand()*[llength $list])}]
}
- Utility: remove an item from a list if it is there
proc remove {list item} {
set idx [lsearch -exact $list $item] return [lreplace $list $idx $idx]
}
- Codify how Monty will present the new set of doors to choose between
proc MontyHallAction {doors car picked} {
set unpicked [remove $doors $picked]
if {$car in $unpicked} {
# Remove a random unpicked door without the car behind it
set carless [remove $unpicked $car]
return [list {*}[remove $carless [pick $carless]] $car]
# Expressed this way so Monty Hall isn't theoretically
# restricted to using 3 doors, though that could be written
# as just: return [list $car]
} else {
# Monty has a real choice now...
return [remove $unpicked [pick $unpicked]]
}
}
- The different strategies you might choose
proc Strategy:Stay {originalPick otherChoices} {
return $originalPick
} proc Strategy:Change {originalPick otherChoices} {
return [pick $otherChoices]
} proc Strategy:PickAnew {originalPick otherChoices} {
return [pick [list $originalPick {*}$otherChoices]]
}
- Codify one round of the game
proc MontyHallGameRound {doors strategy winCounter} {
upvar 1 $winCounter wins
set car [pick $doors]
set picked [pick $doors]
set newDoors [MontyHallAction $doors $car $picked]
set picked [$strategy $picked $newDoors]
# Check for win...
if {$car eq $picked} {
incr wins
}
}
- We're always using three doors
set threeDoors {a b c} set stay 0; set change 0; set anew 0 set total 10000
- Simulate each of the different strategies
for {set i 0} {$i<$total} {incr i} {
MontyHallGameRound $threeDoors Strategy:Stay stay MontyHallGameRound $threeDoors Strategy:Change change MontyHallGameRound $threeDoors Strategy:PickAnew anew
}
- Print the results
puts "Estimate: $stay/$total wins for 'staying' strategy" puts "Estimate: $change/$total wins for 'changing' strategy" puts "Estimate: $anew/$total wins for 'picking anew' strategy"</lang> This might then produce output like
Estimate: 3340/10000 wins for 'staying' strategy Estimate: 6733/10000 wins for 'changing' strategy Estimate: 4960/10000 wins for 'picking anew' strategy
Of course, this challenge could also be tackled by putting up a GUI and letting the user be the source of the randomness. But that's moving away from the letter of the challenge and takes a lot of effort anyway...
Transact SQL
T-SQL for general case: <lang Transact SQL>
BEGIN ------------
create table MONTY_HALL(
NOE int, CAR int, ALTERNATIVE int, ORIGIN int, [KEEP] int, [CHANGE] int, [RANDOM] int )
-- INIT truncate table MONTY_HALL
declare @N int , @i int -- No of Experiments and their counter declare @rooms int , -- number of rooms
@origin int, -- original choice
@car int , -- room with car @alternative int -- alternative room
select @rooms = 3, @N = 100000 , @i = 0
-- EXPERIMENTS LOOP while @i < @N begin
select @car = FLOOR(rand()*@rooms)+1 , @origin = FLOOR(rand()*@rooms)+1 select @alternative = FLOOR(rand()*(@rooms-1))+1 select @alternative = case when @alternative < @origin then @alternative else @alternative + 1 end select @alternative = case when @origin = @car then @alternative else @car end
insert MONTY_HALL select @i,@car,@alternative,@origin,@origin,@alternative,case when rand() < 5e-1 then @origin else @alternative end
select @i = @i + 1
end
-- RESULTS select avg (case when [KEEP] = CAR then 1e0 else 0e0 end )*1e2 as [% OF WINS FOR KEEP],
avg (case when [CHANGE] = CAR then 1e0 else 0e0 end )*1e2 as [% OF WINS FOR CHANGE],
avg (case when [RANDOM] = CAR then 1e0 else 0e0 end )*1e2 as [% OF WINS FOR RANDOM]
from MONTY_HALL
END ------------
</lang>
% OF WINS FOR KEEP % OF WINS FOR CHANGE % OF WINS FOR RANDOM ---------------------- ---------------------- ---------------------- 33.607 66.393 49.938
UNIX Shell
<lang bash>#!/bin/bash
- Simulates the "monty hall" probability paradox and shows results.
- http://en.wikipedia.org/wiki/Monty_Hall_problem
- (should rewrite this in C for faster calculating of huge number of rounds)
- (Hacked up by Éric Tremblay, 07.dec.2010)
num_rounds=10 #default number of rounds num_doors=3 # default number of doors [ "$1" = "" ] || num_rounds=$[$1+0] [ "$2" = "" ] || num_doors=$[$2+0]
nbase=1 # or 0 if we want to see door numbers zero-based num_win=0; num_lose=0
echo "Playing $num_rounds times, with $num_doors doors." [ "$num_doors" -lt 3 ] && {
echo "Hey, there has to be at least 3 doors!!" exit 1
} echo
function one_round() {
winning_door=$[$RANDOM % $num_doors ] player_picks_door=$[$RANDOM % $num_doors ]
# Host leaves this door AND the player's first choice closed, opens all others
# (this WILL loop forever if there is only 1 door)
host_skips_door=$winning_door
while [ "$host_skips_door" = "$player_picks_door" ]; do
#echo -n "(Host looks at door $host_skips_door...) "
host_skips_door=$[$RANDOM % $num_doors]
done
# Output the result of this round
#echo "Round $[$nbase+current_round]: "
echo -n "Player chooses #$[$nbase+$player_picks_door]. "
[ "$num_doors" -ge 10 ] &&
# listing too many door numbers (10 or more) will just clutter the output
echo -n "Host opens all except #$[$nbase+$host_skips_door] and #$[$nbase+$player_picks_door]. " \
|| {
# less than 10 doors, we list them one by one instead of "all except ?? and ??"
echo -n "Host opens"
host_opens=0
while [ "$host_opens" -lt "$num_doors" ]; do
[ "$host_opens" != "$host_skips_door" ] && [ "$host_opens" != "$player_picks_door" ] && \
echo -n " #$[$nbase+$host_opens]"
host_opens=$[$host_opens+1]
done
echo -n " "
}
echo -n "(prize is behind #$[$nbase+$winning_door]) "
echo -n "Switch from $[$nbase+$player_picks_door] to $[$nbase+$host_skips_door]: "
[ "$winning_door" = "$host_skips_door" ] && {
echo "WIN."
num_win=$[num_win+1]
} || {
echo "LOSE."
num_lose=$[num_lose+1]
}
} # end of function one_round
- ok, let's go
current_round=0 while [ "$num_rounds" -gt "$current_round" ]; do
one_round current_round=$[$current_round+1]
done
echo echo "Wins (switch to remaining door): $num_win" echo "Losses (first guess was correct): $num_lose" exit 0</lang> Output of a few runs:
$ ./monty_hall_problem.sh Playing 10 times, with 3 doors. Player chooses #2. Host opens #3 (prize is behind #1) Switch from 2 to 1: WIN. Player chooses #1. Host opens #3 (prize is behind #2) Switch from 1 to 2: WIN. Player chooses #2. Host opens #3 (prize is behind #2) Switch from 2 to 1: LOSE. Player chooses #1. Host opens #2 (prize is behind #1) Switch from 1 to 3: LOSE. Player chooses #2. Host opens #3 (prize is behind #1) Switch from 2 to 1: WIN. Player chooses #2. Host opens #1 (prize is behind #2) Switch from 2 to 3: LOSE. Player chooses #3. Host opens #1 (prize is behind #2) Switch from 3 to 2: WIN. Player chooses #2. Host opens #1 (prize is behind #3) Switch from 2 to 3: WIN. Player chooses #1. Host opens #3 (prize is behind #1) Switch from 1 to 2: LOSE. Player chooses #1. Host opens #2 (prize is behind #3) Switch from 1 to 3: WIN. Wins (switch to remaining door): 6 Losses (first guess was correct): 4 $ ./monty_hall_problem.sh 5 10 Playing 5 times, with 10 doors. Player chooses #1. Host opens all except #10 and #1. (prize is behind #10) Switch from 1 to 10: WIN. Player chooses #7. Host opens all except #8 and #7. (prize is behind #8) Switch from 7 to 8: WIN. Player chooses #6. Host opens all except #1 and #6. (prize is behind #1) Switch from 6 to 1: WIN. Player chooses #8. Host opens all except #3 and #8. (prize is behind #8) Switch from 8 to 3: LOSE. Player chooses #6. Host opens all except #5 and #6. (prize is behind #5) Switch from 6 to 5: WIN. Wins (switch to remaining door): 4 Losses (first guess was correct): 1 $ ./monty_hall_problem.sh 1000 Playing 1000 times, with 3 doors. Player chooses #2. Host opens #1 (prize is behind #2) Switch from 2 to 3: LOSE. Player chooses #3. Host opens #1 (prize is behind #2) Switch from 3 to 2: WIN. [ ... ] Player chooses #1. Host opens #3 (prize is behind #2) Switch from 1 to 2: WIN. Player chooses #3. Host opens #2 (prize is behind #1) Switch from 3 to 1: WIN. Wins (switch to remaining door): 655 Losses (first guess was correct): 345
Ursala
This is the same algorithm as the Perl solution. Generate two lists of 10000 uniformly distributed samples from {1,2,3}, count each match as a win for the staying strategy, and count each non-match as a win for the switching strategy.
<lang Ursala>#import std
- import nat
- import flo
rounds = 10000
car_locations = arc{1,2,3}* iota rounds initial_choices = arc{1,2,3}* iota rounds
staying_wins = length (filter ==) zip(car_locations,initial_choices) switching_wins = length (filter ~=) zip(car_locations,initial_choices)
format = printf/'%0.2f'+ (times\100.+ div+ float~~)\rounds
- show+
main = ~&plrTS/<'stay: ','switch: '> format* <staying_wins,switching_wins></lang> Output will vary slightly for each run due to randomness.
stay: 33.95 switch: 66.05
Vedit macro language
Vedit macro language does not have random number generator, so one is implemented in subroutine RANDOM (the algorithm was taken from ANSI C library). <lang vedit>#90 = Time_Tick // seed for random number generator
- 91 = 3 // random numbers in range 0 to 2
- 1 = 0 // wins for "always stay" strategy
- 2 = 0 // wins for "always switch" strategy
for (#10 = 0; #10 < 10000; #10++) { // 10,000 iterations
Call("RANDOM")
#3 = Return_Value // #3 = winning door
Call("RANDOM")
#4 = Return_Value // #4 = players choice
do {
Call("RANDOM") #5 = Return_Value // #5 = door to open
} while (#5 == #3 || #5 == #4)
if (#3 == #4) { // original choice was correct
#1++
}
if (#3 == 3 - #4 - #5) { // switched choice was correct
#2++
}
} Ins_Text("Staying wins: ") Num_Ins(#1) Ins_Text("Switching wins: ") Num_Ins(#2) return
//-------------------------------------------------------------- // Generate random numbers in range 0 <= Return_Value < #91 // #90 = Seed (0 to 0x7fffffff) // #91 = Scaling (0 to 0xffff)
- RANDOM:
- 92 = 0x7fffffff / 48271
- 93 = 0x7fffffff % 48271
- 90 = (48271 * (#90 % #92) - #93 * (#90 / #92)) & 0x7fffffff
return ((#90 & 0xffff) * #91 / 0x10000)</lang>
Sample output:
Staying winns: 3354 Switching winns: 6646
Wren
<lang ecmascript>import "random" for Random
var montyHall = Fn.new { |games|
var rand = Random.new()
var switchWins = 0
var stayWins = 0
for (i in 1..games) {
var doors = [0] * 3 // all zero (goats) by default
doors[rand.int(3)] = 1 // put car in a random door
var choice = rand.int(3) // choose a door at random
var shown = 0
while (true) {
shown = rand.int(3) // the shown door
if (doors[shown] != 1 && shown != choice) break
}
stayWins = stayWins + doors[choice]
switchWins = switchWins + doors[3-choice-shown]
}
System.print("Simulating %(games) games:")
System.print("Staying wins %(stayWins) times")
System.print("Switching wins %(switchWins) times")
}
montyHall.call(1e6)</lang>
- Output:
Sample output:
Simulating 1000000 games: Staying wins 333970 times Switching wins 666030 times
X++
<lang x++>//Evidence of the Monty Hall solution in Dynamics AX (by Wessel du Plooy - HiGH Software).
int changeWins = 0; int noChangeWins = 0; int attempts; int picked; int reveal; int switchdoor; int doors[];
for (attempts = 0; attempts < 32768; attempts++)
{
doors[1] = 0; //0 is a goat, 1 is a car
doors[2] = 0;
doors[3] = 0;
doors[(xGlobal::randomPositiveInt32() mod 3) + 1] = 1; //put a winner in a random door
picked = (xGlobal::randomPositiveInt32() mod 3) + 1; //pick a door, any door
do
{
reveal = (xGlobal::randomPositiveInt32() mod 3) + 1;
}
while (doors[reveal] == 1 || reveal == picked); //don't show the winner or the choice
if (doors[picked] == 1)
noChangeWins++;
else
changeWins++;
}
print strFmt("Switching wins %1 times.", changeWins);
print strFmt("Staying wins %1 times.", noChangeWins);
pause;
</lang>
Output:
Switching wins 21699 times. Staying wins 11069 times.
XPL0
<lang XPL0>def Games = 10000; \number of games simulated int Game, Wins; include c:\cxpl\codes;
proc Play(Switch); \Play one game int Switch; int Car, Player, Player0, Monty; [Car:= Ran(3); \randomly place car behind a door Player0:= Ran(3); \player randomly chooses a door repeat Monty:= Ran(3); \Monty opens door revealing a goat until Monty # Car and Monty # Player0; if Switch then \player switches to remaining door
repeat Player:= Ran(3);
until Player # Player0 and Player # Monty
else Player:= Player0; \player sticks with original door if Player = Car then Wins:= Wins+1; ];
[Format(2,1); Text(0, "Not switching doors wins car in "); Wins:= 0; for Game:= 0 to Games-1 do Play(false); RlOut(0, float(Wins)/float(Games)*100.0); Text(0, "% of games.^M^J");
Text(0, "But switching doors wins car in "); Wins:= 0; for Game:= 0 to Games-1 do Play(true); RlOut(0, float(Wins)/float(Games)*100.0); Text(0, "% of games.^M^J"); ]</lang>
Example output:
Not switching doors wins car in 33.7% of games. But switching doors wins car in 66.7% of games.
Yabasic
<lang yabasic> numTiradas = 1000000
for i = 1 to numTiradas pta_coche = int(ran(3)) + 1 pta_elegida = int(ran(3)) + 1 if pta_coche <> pta_elegida then pta_montys = 6 - pta_coche - pta_elegida else repeat pta_montys = int(ran(3)) + 1 until pta_montys <> pta_coche end if // manteenr elección if pta_coche = pta_elegida then permanece = permanece + 1 : fi // cambiar elección if pta_coche = 6 - pta_montys - pta_elegida then cambia = cambia + 1 : fi next i
print "Si mantiene su eleccion, tiene un ", permanece / numTiradas * 100, "% de probabilidades de ganar." print "Si cambia, tiene un ", cambia / numTiradas * 100, "% de probabilidades de ganar." end </lang>
zkl
<lang zkl>const games=0d100_000;
reg switcherWins=0, keeperWins=0, shown=0; do(games){
doors := L(0,0,0);
doors[(0).random(3)] = 1; // Set which one has the car
choice := (0).random(3); // Choose a door
while(1){ shown = (0).random(3);
if (not (shown == choice or doors[shown] == 1)) break; }
switcherWins += doors[3 - choice - shown];
keeperWins += doors[choice];
}
"Switcher Wins: %,d (%3.2f%%)".fmt(
switcherWins, switcherWins.toFloat() / games * 100).println();
"Keeper Wins: %,d (%3.2f%%)".fmt(
keeperWins, keeperWins.toFloat() / games * 100).println();</lang>
- Output:
Switcher Wins: 66,730 (66.73%) Keeper Wins: 33,270 (33.27%)
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