Dice game probabilities
You are encouraged to solve this task according to the task description, using any language you may know.
Two players have a set of dice each. The first player has nine dice with four faces each, with numbers one to four. The second player has six normal dice with six faces each, each face has the usual numbers from one to six.
They roll their dice and sum the totals of the faces. The player with the highest total wins (it's a draw if the totals are the same). What's the probability of the first player beating the second player?
Later the two players use a different set of dice each. Now the first player has five dice with ten faces each, and the second player has six dice with seven faces each. Now what's the probability of the first player beating the second player?
This task was adapted from the Project Euler Problem n.205: https://projecteuler.net/problem=205
11l
<lang 11l>F throw_die(n_sides, n_dice, s, [Int] &counts)
I n_dice == 0
counts[s]++
R
L(i) 1..n_sides
throw_die(n_sides, n_dice - 1, s + i, counts)
F beating_probability(n_sides1, n_dice1,
n_sides2, n_dice2) V len1 = (n_sides1 + 1) * n_dice1 V C1 = [0] * len1 throw_die(n_sides1, n_dice1, 0, &C1)
V len2 = (n_sides2 + 1) * n_dice2 V C2 = [0] * len2 throw_die(n_sides2, n_dice2, 0, &C2)
Float p12 = (n_sides1 ^ n_dice1) * (n_sides2 ^ n_dice2)
V tot = 0.0
L(i) 0 .< len1
L(j) 0 .< min(i, len2)
tot += Float(C1[i]) * C2[j] / p12
R tot
print(‘#.16’.format(beating_probability(4, 9, 6, 6))) print(‘#.16’.format(beating_probability(10, 5, 7, 6)))</lang>
- Output:
0.5731440767829814 0.6427886287176272
Action!
<lang Action!>INCLUDE "D2:REAL.ACT" ;from the Action! Tool Kit
BYTE FUNC Roll(BYTE sides,dices)
BYTE i,sum
sum=0 FOR i=1 TO dices DO sum==+Rand(sides)+1 OD
RETURN (sum)
PROC Test(BYTE sides1,dices1,sides2,dices2)
INT i,count=[10000],sum1,sum2,wins1,wins2,draws REAL r1,r2,p
wins1=0 wins2=0 draws=0
FOR i=1 TO count
DO
sum1=Roll(sides1,dices1)
sum2=Roll(sides2,dices2)
IF sum1>sum2 THEN
wins1==+1
ELSEIF sum1<sum2 THEN
wins2==+1
ELSE
draws==+1
FI
OD
IntToReal(wins1,r1) IntToReal(wins2,r2) RealDiv(r2,r1,p)
PrintF("Tested %I times%E",count)
PrintF("Player 1 with %B dice of %B sides%E",dices1,sides1)
PrintF("Player 2 with %B dice of %B sides%E",dices2,sides2)
PrintF("Player 1 wins %I times%E",wins1)
PrintF("Player 2 wins %I times%E",wins2)
PrintF("Draw %I times%E",draws)
Print("Player 1 beating Player 2 probability:")
PrintRE(p)
PutE()
RETURN
PROC Main()
Put(125) PutE() ;clear the screen
Test(4,9,6,6) Test(10,5,7,6)
RETURN</lang>
- Output:
Screenshot from Atari 8-bit computer
Tested 10000 times Player 1 with 9 dice of 4 sides Player 2 with 6 dice of 6 sides Player 1 wins 5770 times Player 2 wins 3493 times Draw 737 times Player 1 beating Player 2 probability: .6053726169 Tested 10000 times Player 1 with 5 dice of 10 sides Player 2 with 6 dice of 7 sides Player 1 wins 6417 times Player 2 wins 3140 times Draw 443 times Player 1 beating Player 2 probability: .4893252298
Ada
<lang Ada>with Ada.Text_IO; use Ada.Text_IO; with Ada.Numerics.Discrete_Random;
procedure Main is
package real_io is new Float_IO (Long_Float); use real_io;
type Dice is record
Faces : Positive;
Num_Dice : Positive;
end record;
procedure Roll_Dice (The_Dice : in Dice; Count : out Natural) is
subtype Faces is Integer range 1 .. The_Dice.Faces;
package Die_Random is new Ada.Numerics.Discrete_Random (Faces);
use Die_Random;
Seed : Generator;
begin
Reset (Seed);
Count := 0;
for I in 1 .. The_Dice.Num_Dice loop
Count := Count + Random (Seed);
end loop;
end Roll_Dice;
function Win_Prob
(Dice_1 : Dice; Dice_2 : Dice; Tries : Positive) return Long_Float
is
Count_1 : Natural := 0;
Count_2 : Natural := 0;
Count_1_Wins : Natural := 0;
begin
for I in 1 .. Tries loop
Roll_Dice (Dice_1, Count_1);
Roll_Dice (Dice_2, Count_2);
if Count_1 > Count_2 then
Count_1_Wins := Count_1_Wins + 1;
end if;
end loop;
return Long_Float (Count_1_Wins) / Long_Float (Tries);
end Win_Prob;
D1 : Dice := (Faces => 4, Num_Dice => 9); D2 : Dice := (Faces => 6, Num_Dice => 6); D3 : Dice := (Faces => 10, Num_Dice => 5); D4 : Dice := (Faces => 7, Num_Dice => 6);
P1 : Long_Float := Win_Prob (D1, D2, 1_000_000); P2 : Long_Float := Win_Prob (D3, D4, 1_000_000);
begin
Put ("Dice D1 wins = ");
Put (Item => P1, Fore => 1, Aft => 7, Exp => 0);
New_Line;
Put ("Dice D2 wins = ");
Put (Item => P2, Fore => 1, Aft => 7, Exp => 0);
New_Line;
end Main; </lang>
- Output:
Dice D1 wins = 0.5727800 Dice D2 wins = 0.6427660
BASIC
BASIC256
<lang BASIC256>dado1 = 9: lado1 = 4 dado2 = 6: lado2 = 6 total1 = 0: total2 = 0
for i = 0 to 1
for cont = 1 to 100000
jugador1 = lanzamiento(dado1, lado1)
jugador2 = lanzamiento(dado2, lado2)
if jugador1 > jugador2 then
total1 = total1 + 1
else
if jugador1 <> jugador2 then total2 = total2 + 1
endif
next cont
print "Lanzado el dado "; (cont - 1); " veces" print "jugador1 con "; dado1; " dados de "; lado1; " lados" print "jugador2 con "; dado2; " dados de "; lado2; " lados" print "Total victorias jugador1 = "; total1; " => "; left(string(total2 / total1), 9) print "Total victorias jugador2 = "; total2 print (cont - 1) - (total1 + total2); " empates" + chr(10)
dado1 = 5: lado1 = 10 dado2 = 6: lado2 = 7 total1 = 0: total2 = 0
next i end
function lanzamiento(dado, lado)
total = 0
for lanza = 1 to dado
total = total + int(rand * lado) + 1
next lanza
return total
end function</lang>
- Output:
Igual que la entrada de Yabasic.
FreeBASIC
<lang freebasic>Dim As Integer lado, jugador1, jugador2, total1, total2, cont, i Dim As Integer dado1 = 9, lado1 = 4 Dim As Integer dado2 = 6, lado2 = 6
Randomize Timer
Function Lanzamiento(dado As Integer, lado As Integer) As Integer
Dim As Short lanza, total
For lanza = 1 To dado
total += Int(Rnd * lado) + 1
Next lanza
Return total
End Function
For i = 0 To 1
For cont = 1 To 100000
jugador1 = Lanzamiento(dado1, lado1)
jugador2 = Lanzamiento(dado2, lado2)
If jugador1 > jugador2 Then
total1 += 1
Elseif jugador1 <> jugador2 Then
total2 += 1
End If
Next cont
Print Using "Lanzado el dado & veces"; (cont - 1)
Print "jugador1 con"; dado1; " dados de"; lado1; " lados"
Print "jugador2 con"; dado2; " dados de"; lado2; " lados"
Print Using "Total victorias jugador1 = & => #.#######"; total1; (total2 / total1)
Print "Total victorias jugador2 ="; total2
Print (cont - 1) - (total1 + total2); !" empates\n"
dado1 = 5: lado1 = 10
dado2 = 6: lado2 = 7
total1 = 0: total2 = 0
Next i
Sleep</lang>
- Output:
Lanzado el dado 100000 veces jugador1 con 9 dados de 4 lados jugador2 con 6 dados de 6 lados Total victorias jugador1 = 57274 => 0.6237211 Total victorias jugador2 = 35723 7003 empates Lanzado el dado 100000 veces jugador1 con 5 dados de 10 lados jugador2 con 6 dados de 7 lados Total victorias jugador1 = 64093 => 0.4893826 Total victorias jugador2 = 31366 4541 empates
Yabasic
<lang Yabasic>dado1 = 9: lado1 = 4 dado2 = 6: lado2 = 6 total1 = 0: total2 = 0
for i = 0 to 1
for cont = 1 to 100000
jugador1 = lanzamiento(dado1, lado1)
jugador2 = lanzamiento(dado2, lado2)
if jugador1 > jugador2 then
total1 = total1 + 1
elseif jugador1 <> jugador2 then
total2 = total2 + 1
endif
next cont
print "Lanzado el dado ", (cont - 1), " veces"
print "jugador1 con ", dado1, " dados de ", lado1, " lados"
print "jugador2 con ", dado2, " dados de ", lado2, " lados"
print "Total victorias jugador1 = ", total1, " => ", (total2 / total1)
print "Total victorias jugador2 = ", total2
print (cont - 1) - (total1 + total2), " empates\n"
dado1 = 5: lado1 = 10
dado2 = 6: lado2 = 7
total1 = 0: total2 = 0
next i
sub lanzamiento(dado, lado)
local lanza, total
for lanza = 1 to dado
total = total + int(ran(lado)) + 1
next lanza
return total
end sub</lang>
- Output:
Igual que la entrada de FreeBASIC.
C
<lang c>#include <stdio.h>
- include <stdint.h>
typedef uint32_t uint; typedef uint64_t ulong;
ulong ipow(const uint x, const uint y) {
ulong result = 1;
for (uint i = 1; i <= y; i++)
result *= x;
return result;
}
uint min(const uint x, const uint y) {
return (x < y) ? x : y;
}
void throw_die(const uint n_sides, const uint n_dice, const uint s, uint counts[]) {
if (n_dice == 0) {
counts[s]++;
return;
}
for (uint i = 1; i < n_sides + 1; i++)
throw_die(n_sides, n_dice - 1, s + i, counts);
}
double beating_probability(const uint n_sides1, const uint n_dice1,
const uint n_sides2, const uint n_dice2) {
const uint len1 = (n_sides1 + 1) * n_dice1;
uint C1[len1];
for (uint i = 0; i < len1; i++)
C1[i] = 0;
throw_die(n_sides1, n_dice1, 0, C1);
const uint len2 = (n_sides2 + 1) * n_dice2;
uint C2[len2];
for (uint j = 0; j < len2; j++)
C2[j] = 0;
throw_die(n_sides2, n_dice2, 0, C2);
const double p12 = (double)(ipow(n_sides1, n_dice1) * ipow(n_sides2, n_dice2));
double tot = 0;
for (uint i = 0; i < len1; i++)
for (uint j = 0; j < min(i, len2); j++)
tot += (double)C1[i] * C2[j] / p12;
return tot;
}
int main() {
printf("%1.16f\n", beating_probability(4, 9, 6, 6));
printf("%1.16f\n", beating_probability(10, 5, 7, 6));
return 0;
}</lang>
- Output:
0.5731440767829801 0.6427886287176260
C++
<lang cpp>#include <cmath>
- include <cstdint>
- include <iomanip>
- include <iostream>
- include <map>
// Returns map from sum of faces to number of ways it can happen std::map<uint32_t, uint32_t> get_totals(uint32_t dice, uint32_t faces) {
std::map<uint32_t, uint32_t> result;
for (uint32_t i = 1; i <= faces; ++i)
result.emplace(i, 1);
for (uint32_t d = 2; d <= dice; ++d) {
std::map<uint32_t, uint32_t> tmp;
for (const auto& p : result) {
for (uint32_t i = 1; i <= faces; ++i)
tmp[p.first + i] += p.second;
}
tmp.swap(result);
}
return result;
}
double probability(uint32_t dice1, uint32_t faces1, uint32_t dice2, uint32_t faces2) {
auto totals1 = get_totals(dice1, faces1);
auto totals2 = get_totals(dice2, faces2);
double wins = 0;
for (const auto& p1 : totals1) {
for (const auto& p2 : totals2) {
if (p2.first >= p1.first)
break;
wins += p1.second * p2.second;
}
}
double total = std::pow(faces1, dice1) * std::pow(faces2, dice2);
return wins/total;
}
int main() {
std::cout << std::setprecision(10); std::cout << probability(9, 4, 6, 6) << '\n'; std::cout << probability(5, 10, 6, 7) << '\n'; return 0;
}</lang>
- Output:
0.5731440768 0.6427886287
D
version 1
<lang d>import std.stdio, std.range, std.algorithm;
void throwDie(in uint nSides, in uint nDice, in uint s, uint[] counts) pure nothrow @safe @nogc {
if (nDice == 0) {
counts[s]++;
return;
}
foreach (immutable i; 1 .. nSides + 1)
throwDie(nSides, nDice - 1, s + i, counts);
}
real beatingProbability(uint nSides1, uint nDice1,
uint nSides2, uint nDice2)()
pure nothrow @safe /*@nogc*/ {
uint[(nSides1 + 1) * nDice1] C1; throwDie(nSides1, nDice1, 0, C1);
uint[(nSides2 + 1) * nDice2] C2; throwDie(nSides2, nDice2, 0, C2);
immutable p12 = real((ulong(nSides1) ^^ nDice1) *
(ulong(nSides2) ^^ nDice2));
return cartesianProduct(C1[].enumerate, C2[].enumerate)
.filter!(p => p[0][0] > p[1][0])
.map!(p => real(p[0][1]) * p[1][1] / p12)
.sum;
}
void main() @safe {
writefln("%1.16f", beatingProbability!(4, 9, 6, 6));
writefln("%1.16f", beatingProbability!(10, 5, 7, 6));
}</lang>
- Output:
0.5731440767829801 0.6427886287176262
version 2 (Faster Alternative Version)
<lang d>import std.stdio, std.range, std.algorithm;
ulong[] combos(R)(R sides, in uint n) pure nothrow @safe if (isForwardRange!R) {
if (sides.empty)
return null;
if (!n)
return [1];
auto ret = new typeof(return)(reduce!max(sides[0], sides[1 .. $]) * n + 1);
foreach (immutable i, immutable v; enumerate(combos(sides, n - 1))) {
if (!v)
continue;
foreach (immutable s; sides)
ret[i + s] += v;
}
return ret;
}
real winning(R)(R sides1, in uint n1, R sides2, in uint n2) pure nothrow @safe if (isForwardRange!R) {
static void accumulate(T)(T[] arr) pure nothrow @safe @nogc {
foreach (immutable i; 1 .. arr.length)
arr[i] += arr[i - 1];
}
immutable p1 = combos(sides1, n1);
auto p2 = combos(sides2, n2);
immutable s = p1.sum * p2.sum;
accumulate(p2);
ulong win = 0; // 'win' is 1 beating 2.
foreach (immutable i, immutable x1; p1.dropOne.enumerate)
win += x1 * p2[min(i, $ - 1)];
return win / real(s);
}
void main() @safe {
writefln("%1.16f", winning(iota(1u, 5u), 9, iota(1u, 7u), 6));
writefln("%1.16f", winning(iota(1u, 11u), 5, iota(1u, 8u), 6));
}</lang>
- Output:
0.5731440767829801 0.6427886287176262
Factor
<lang factor>USING: dice generalizations kernel math prettyprint sequences ; IN: rosetta-code.dice-probabilities
- winning-prob ( a b c d -- p )
[ [ random-roll ] 2bi@ > ] 4 ncurry [ 100000 ] dip replicate [ [ t = ] count ] [ length ] bi /f ;
9 4 6 6 winning-prob 5 10 6 7 winning-prob [ . ] bi@</lang>
- Output:
0.57199 0.64174
Gambas
Click this link to run this code <lang gambas>' Gambas module file
Public Sub Main() Dim iSides, iPlayer1, iPlayer2, iTotal1, iTotal2, iCount, iCount0 As Integer Dim iDice1 As Integer = 9 Dim iDice2 As Integer = 6 Dim iSides1 As Integer = 4 Dim iSides2 As Integer = 6
Randomize
For iCount0 = 0 To 1
For iCount = 1 To 100000
iPlayer1 = Roll(iDice1, iSides1)
iPlayer2 = Roll(iDice2, iSides2)
If iPlayer1 > iPlayer2 Then
iTotal1 += 1
Else If iPlayer1 <> iPlayer2 Then
iTotal2 += 1
Endif
Next
Print "Tested " & Str(iCount - 1) & " times" Print "Player1 with " & Str(iDice1) & " dice of " & Str(iSides1) & " sides" Print "Player2 with " & Str(iDice2) & " dice of " & Str(iSides2) & " sides" Print "Total wins Player1 = " & Str(iTotal1) & " - " & Str(iTotal2 / iTotal1) Print "Total wins Player2 = " & Str(iTotal2) Print Str((iCount - 1) - (iTotal1 + iTotal2)) & " draws" & gb.NewLine
iDice1 = 5 iDice2 = 6 iSides1 = 10 iSides2 = 7 iTotal1 = 0 iTotal2 = 0
Next
End
Public Sub Roll(iDice As Integer, iSides As Integer) As Integer Dim iCount, iTotal As Short
For iCount = 1 To iDice
iTotal += Rand(1, iSides)
Next
Return iTotal
End</lang> Output:
Tested 100000 times Player1 with 9 dice of 4 sides Player2 with 6 dice of 6 sides Total wins Player1 = 56980 - 0.62823797823798 Total wins Player2 = 35797 7223 draws Tested 100000 times Player1 with 5 dice of 10 sides Player2 with 6 dice of 7 sides Total wins Player1 = 64276 - 0.48548447320928 Total wins Player2 = 31205 4519 draws
Go
<lang go>package main
import(
"math" "fmt"
)
func minOf(x, y uint) uint {
if x < y {
return x
}
return y
}
func throwDie(nSides, nDice, s uint, counts []uint) {
if nDice == 0 {
counts[s]++
return
}
for i := uint(1); i <= nSides; i++ {
throwDie(nSides, nDice - 1, s + i, counts)
}
}
func beatingProbability(nSides1, nDice1, nSides2, nDice2 uint) float64 {
len1 := (nSides1 + 1) * nDice1 c1 := make([]uint, len1) // all elements zero by default throwDie(nSides1, nDice1, 0, c1)
len2 := (nSides2 + 1) * nDice2
c2 := make([]uint, len2)
throwDie(nSides2, nDice2, 0, c2)
p12 := math.Pow(float64(nSides1), float64(nDice1)) *
math.Pow(float64(nSides2), float64(nDice2))
tot := 0.0
for i := uint(0); i < len1; i++ {
for j := uint(0); j < minOf(i, len2); j++ {
tot += float64(c1[i] * c2[j]) / p12
}
}
return tot
}
func main() {
fmt.Println(beatingProbability(4, 9, 6, 6)) fmt.Println(beatingProbability(10, 5, 7, 6))
}</lang>
- Output:
0.5731440767829815 0.6427886287176273
More idiomatic go:
<lang go>package main
import ( "fmt" "math/rand" )
type set struct { n, s int }
func (s set) roll() (sum int) { for i := 0; i < s.n; i++ { sum += rand.Intn(s.s) + 1 } return }
func (s set) beats(o set, n int) (p float64) { for i := 0; i < n; i++ { if s.roll() > o.roll() { p = p + 1.0 } } p = p / float64(n) return }
func main() { fmt.Println(set{9, 4}.beats(set{6, 6}, 1000)) fmt.Println(set{5, 10}.beats(set{6, 7}, 1000)) }</lang>
- Output:
0.576 0.639
Haskell
<lang haskell>import Control.Monad (replicateM) import Data.List (group, sort)
succeeds :: (Int, Int) -> (Int, Int) -> Double succeeds p1 p2 =
sum
[ realToFrac (c1 * c2) / totalOutcomes
| (s1, c1) <- countSums p1
, (s2, c2) <- countSums p2
, s1 > s2 ]
where
totalOutcomes = realToFrac $ uncurry (^) p1 * uncurry (^) p2
countSums (nFaces, nDice) = f [1 .. nFaces]
where
f =
fmap (((,) . head) <*> (pred . length)) .
group . sort . fmap sum . replicateM nDice
main :: IO () main = do
print $ (4, 9) `succeeds` (6, 6) print $ (10, 5) `succeeds` (7, 6)</lang>
- Output:
0.5731440767829815 0.6427886287176273
J
Solution: <lang J>gen_dict =: (({. , #)/.~@:,@:(+/&>)@:{@:(# <@:>:@:i.)~ ; ^)&x:
beating_probability =: dyad define
'C0 P0' =. gen_dict/ x
'C1 P1' =. gen_dict/ y
(C0 +/@:,@:(>/&:({."1) * */&:({:"1)) C1) % (P0 * P1)
)</lang> Example Usage: <lang J> 10 5 (;x:inv)@:beating_probability 7 6 ┌─────────────────────┬────────┐ │3781171969r5882450000│0.642789│ └─────────────────────┴────────┘
4 9 (;x:inv)@:beating_probability 6 6
┌─────────────────┬────────┐
│48679795r84934656│0.573144│
└─────────────────┴────────┘</lang>
gen_dict explanation:
gen_dict is akin to gen_dict in the python solution and
returns a table and total number of combinations, order matters.
The table has 2 columns. The first column is the pip count on all dice,
the second column is the number of ways this many pips can occur.
({. , #)/.~ make a vector having items head and tally of each group of like items in this case pip count and occurrences operating on
, raveled data (a vector) made of
+/&> the sum of each of the
{ Cartesian products of the
(# <@:>:@:i.) equi-probable die values
&x: but first use extended integers
; ^ links the total possibilities to the result.
The verb beating_probability is akin to the python solution function having same name.
C0 >/&:({."1) C1 is a binary table where the pips of first player exceed pips of second player. "Make a greater than table but first take the head of each item."
C0 */&:({:"1) C1 is the corresponding table of occurrences
* naturally we multiply the two tables (atom by atom, not a matrix product)
+/@:,@: sum the raveled table
% (P0 * P1) after which divide by the all possible rolls.
Java
<lang java>import java.util.Random;
public class Dice{ private static int roll(int nDice, int nSides){ int sum = 0; Random rand = new Random(); for(int i = 0; i < nDice; i++){ sum += rand.nextInt(nSides) + 1; } return sum; }
private static int diceGame(int p1Dice, int p1Sides, int p2Dice, int p2Sides, int rolls){ int p1Wins = 0; for(int i = 0; i < rolls; i++){ int p1Roll = roll(p1Dice, p1Sides); int p2Roll = roll(p2Dice, p2Sides); if(p1Roll > p2Roll) p1Wins++; } return p1Wins; }
public static void main(String[] args){ int p1Dice = 9; int p1Sides = 4; int p2Dice = 6; int p2Sides = 6; int rolls = 10000; int p1Wins = diceGame(p1Dice, p1Sides, p2Dice, p2Sides, rolls); System.out.println(rolls + " rolls, p1 = " + p1Dice + "d" + p1Sides + ", p2 = " + p2Dice + "d" + p2Sides); System.out.println("p1 wins " + (100.0 * p1Wins / rolls) + "% of the time");
System.out.println();
p1Dice = 5; p1Sides = 10; p2Dice = 6; p2Sides = 7; rolls = 10000; p1Wins = diceGame(p1Dice, p1Sides, p2Dice, p2Sides, rolls); System.out.println(rolls + " rolls, p1 = " + p1Dice + "d" + p1Sides + ", p2 = " + p2Dice + "d" + p2Sides); System.out.println("p1 wins " + (100.0 * p1Wins / rolls) + "% of the time");
System.out.println();
p1Dice = 9; p1Sides = 4; p2Dice = 6; p2Sides = 6; rolls = 1000000; p1Wins = diceGame(p1Dice, p1Sides, p2Dice, p2Sides, rolls); System.out.println(rolls + " rolls, p1 = " + p1Dice + "d" + p1Sides + ", p2 = " + p2Dice + "d" + p2Sides); System.out.println("p1 wins " + (100.0 * p1Wins / rolls) + "% of the time");
System.out.println();
p1Dice = 5; p1Sides = 10; p2Dice = 6; p2Sides = 7; rolls = 1000000; p1Wins = diceGame(p1Dice, p1Sides, p2Dice, p2Sides, rolls); System.out.println(rolls + " rolls, p1 = " + p1Dice + "d" + p1Sides + ", p2 = " + p2Dice + "d" + p2Sides); System.out.println("p1 wins " + (100.0 * p1Wins / rolls) + "% of the time"); } }</lang>
- Output:
10000 rolls, p1 = 9d4, p2 = 6d6 p1 wins 57.56% of the time 10000 rolls, p1 = 5d10, p2 = 6d7 p1 wins 64.28% of the time 1000000 rolls, p1 = 9d4, p2 = 6d6 p1 wins 57.3563% of the time 1000000 rolls, p1 = 5d10, p2 = 6d7 p1 wins 64.279% of the time
JacaScript
simulating
<lang javascript> let Player = function(dice, faces) {
this.dice = dice;
this.faces = faces;
this.roll = function() {
let results = [];
for (let x = 0; x < dice; x++)
results.push(Math.floor(Math.random() * faces +1));
return eval(results.join('+'));
}
}
function contest(player1, player2, rounds) {
let res = [0, 0, 0];
for (let x = 1; x <= rounds; x++) {
let a = player1.roll(),
b = player2.roll();
switch (true) {
case (a > b): res[0]++; break;
case (a < b): res[1]++; break;
case (a == b): res[2]++; break;
}
}
document.write(`
Player 1 (${player1.dice} × d${player1.faces}): ${res[0]} wins
Player 2 (${player2.dice} × d${player2.faces}): ${res[1]} wins
Draws: ${res[2]}
Chances for Player 1 to win:
~${Math.round(res[0] / eval(res.join('+')) * 100)} %
`);
}
let p1, p2;
p1 = new Player(9, 4), p2 = new Player(6, 6); contest(p1, p2, 1e6);
p1 = new Player(5, 10); p2 = new Player(6, 7); contest(p1, p2, 1e6); </lang>
- Output:
Player 1 (9 × d4): 572753 wins Player 2 (6 × d6): 356478 wins Draws: 70769 Chances for Player 1 to win: ~57 % Player 1 (5 × d10): 643127 wins Player 2 (6 × d7): 312151 wins Draws: 44722 Chances for Player 1 to win: ~64 %
jq
Works with gojq, the Go implementation of jq <lang jq># To take advantage of gojq's arbitrary-precision integer arithmetic: def power($b): . as $in | reduce range(0;$b) as $i (1; . * $in);
- Input: an array (aka: counts)
def throwDie($nSides; $nDice; $s):
if $nDice == 0 then .[$s] += 1 else reduce range(1; $nSides + 1) as $i (.; throwDie($nSides; $nDice-1; $s + $i) ) end ;
def beatingProbability(nSides1; nDice1; nSides2; nDice2):
def a: [range(0; .) | 0];
((nSides1 + 1) * nDice1) as $len1 | ($len1 | a | throwDie(nSides1; nDice1; 0)) as $c1
| ((nSides2 + 1) * nDice2) as $len2 | ($len2 | a | throwDie(nSides2; nDice2; 0)) as $c2
|((nSides1|power(nDice1)) * (nSides2|power(nDice2))) as $p12
| reduce range(0; $len1) as $i (0;
reduce range(0; [$i, $len2] | min) as $j (.;
. + ($c1[$i] * $c2[$j] / $p12) ) ) ;
beatingProbability(4; 9; 6; 6), beatingProbability(10; 5; 7; 6)</lang>
- Output:
0.5731440767829815 0.6427886287176273
Julia
<lang julia>play(ndices::Integer, nfaces::Integer) = (nfaces, ndices) ∋ 0 ? 0 : sum(rand(1:nfaces) for i in 1:ndices)
simulate(d1::Integer, f1::Integer, d2::Integer, f2::Integer; nrep::Integer=1_000_000) =
mean(play(d1, f1) > play(d2, f2) for _ in 1:nrep)
println("\nPlayer 1: 9 dices, 4 faces\nPlayer 2: 6 dices, 6 faces\nP(Player1 wins) = ", simulate(9, 4, 6, 6)) println("\nPlayer 1: 5 dices, 10 faces\nPlayer 2: 6 dices, 7 faces\nP(Player1 wins) = ", simulate(5, 10, 6, 7))</lang>
- Output:
Player 1: 9 dices, 4 faces Player 2: 6 dices, 6 faces P(Player1 wins) = 0.572805 Player 1: 5 dices, 10 faces Player 2: 6 dices, 7 faces P(Player1 wins) = 0.642727
Kotlin
<lang scala>// version 1.1.2
fun throwDie(nSides: Int, nDice: Int, s: Int, counts: IntArray) {
if (nDice == 0) {
counts[s]++
return
}
for (i in 1..nSides) throwDie(nSides, nDice - 1, s + i, counts)
}
fun beatingProbability(nSides1: Int, nDice1: Int, nSides2: Int, nDice2: Int): Double {
val len1 = (nSides1 + 1) * nDice1 val c1 = IntArray(len1) // all elements zero by default throwDie(nSides1, nDice1, 0, c1)
val len2 = (nSides2 + 1) * nDice2 val c2 = IntArray(len2) throwDie(nSides2, nDice2, 0, c2)
val p12 = Math.pow(nSides1.toDouble(), nDice1.toDouble()) *
Math.pow(nSides2.toDouble(), nDice2.toDouble())
var tot = 0.0
for (i in 0 until len1) {
for (j in 0 until minOf(i, len2)) {
tot += c1[i] * c2[j] / p12
}
}
return tot
}
fun main(args: Array<String>) {
println(beatingProbability(4, 9, 6, 6)) println(beatingProbability(10, 5, 7, 6))
}</lang>
- Output:
0.5731440767829815 0.6427886287176273
Lua
Simulated
<lang lua> local function simu(ndice1, nsides1, ndice2, nsides2)
local function roll(ndice, nsides)
local result = 0;
for i = 1, ndice do
result = result + math.random(nsides)
end
return result
end
local wins, coms = 0, 1000000
for i = 1, coms do
local roll1 = roll(ndice1, nsides1)
local roll2 = roll(ndice2, nsides2)
if (roll1 > roll2) then
wins = wins + 1
end
end
print("simulated: p1 = "..ndice1.."d"..nsides1..", p2 = "..ndice2.."d"..nsides2..", prob = "..wins.." / "..coms.." = "..(wins/coms))
end
simu(9, 4, 6, 6) simu(5, 10, 6, 7) </lang>
- Output:
simulated: p1 = 9d4, p2 = 6d6, prob = 573372 / 1000000 = 0.573372 simulated: p1 = 5d10, p2 = 6d7, prob = 642339 / 1000000 = 0.642339
Computed
<lang lua> local function comp(ndice1, nsides1, ndice2, nsides2)
local function throws(ndice, nsides)
local sums = {}
for i = 1, ndice*nsides do
sums[i] = 0
end
local function throw(ndice, nsides, s)
if (ndice==0) then
sums[s] = sums[s] + 1
else
for i = 1, nsides do
throw(ndice-1, nsides, s+i)
end
end
return sums
end
return throw(ndice, nsides, 0)
end
local p1 = throws(ndice1, nsides1)
local p2 = throws(ndice2, nsides2)
local wins, coms = 0, nsides1^ndice1 * nsides2^ndice2
for k1,v1 in pairs(p1) do
for k2,v2 in pairs(p2) do
if (k1 > k2) then
wins = wins + v1 * v2
end
end
end
print("computed: p1 = "..ndice1.."d"..nsides1..", p2 = "..ndice2.."d"..nsides2..", prob = "..wins.." / "..coms.." = "..(wins/coms))
end
comp(9, 4, 6, 6) comp(5, 10, 6, 7) </lang>
- Output:
computed: p1 = 9d4, p2 = 6d6, prob = 7009890480 / 12230590464 = 0.57314407678298 computed: p1 = 5d10, p2 = 6d7, prob = 7562343938 / 11764900000 = 0.64278862871763
Mathematica / Wolfram Language
<lang Mathematica>ClearAll[GetProbability] GetProbability[{dice1_List, n_Integer}, {dice2_List, m_Integer}] :=
Module[{a, b, lena, lenb},
a = Tuples[dice1, n];
a = Plus @@@ a;
lena = a // Length;
a = Tally[a];
aAll, 2 /= lena;
b = Tuples[dice2, m];
b = Plus @@@ b;
lenb = b // Length;
b = Tally[b];
bAll, 2 /= lenb;
Total[If[#1, 1 > #2, 1, #1, 2 #2, 2, 0] & /@
Tuples[{a, b}]]
]
GetProbability[{Range[4], 9}, {Range[6], 6}] GetProbability[{Range[10], 5}, {Range[7], 6}]</lang>
- Output:
48679795/84934656 3781171969/5882450000
Nim
<lang Nim>import bignum from math import sum
proc counts(ndices, nsides: int): seq[int] =
result.setlen(ndices * nsides + 1)
for i in 1..nsides:
result[i] = 1
for i in 1..<ndices:
var c = newSeq[int](result.len)
for sum in i..(i * nsides):
for val in 1..nsides:
inc c[sum + val], result[sum]
result.shallowCopy(c)
proc probabilities(counts: seq[int]): seq[Rat] =
result.setLen(counts.len) let total = sum(counts) for i, n in counts: result[i] = newRat(n, total)
proc beatingProbability(ndices1, nsides1, ndices2, nsides2: int): Rat =
let counts1 = counts(ndices1, nsides1) let counts2 = counts(ndices2, nsides2) var p1 = counts1.probabilities() var p2 = counts2.probabilities()
result = newRat(0)
for sum1 in ndices1..p1.high:
var p = newRat(0)
for sum2 in ndices2..min(sum1 - 1, p2.high):
p += p2[sum2]
result += p1[sum1] * p
echo beatingProbability(9, 4, 6, 6).toFloat echo beatingProbability(5, 10, 6, 7).toFloat</lang>
- Output:
0.5731440767829801 0.6427886287176261
ooRexx
Algorithm
<lang oorexx>Numeric Digits 30 Call test '9 4 6 6' Call test '5 10 6 7' Exit test: Parse Arg w1 s1 w2 s2 p1.=0 p2.=0 Call pp 1,w1,s1,p1.,p2. Call pp 2,w2,s2,p1.,p2. p2low.=0 Do x=w1 To w1*s1
Do y=0 To x-1 p2low.x+=p2.y End End
pwin1=0 Do x=w1 To w1*s1
pwin1+=p1.x*p2low.x End
Say 'Player 1 has' w1 'dice with' s1 'sides each' Say 'Player 2 has' w2 'dice with' s2 'sides each' Say 'Probability for player 1 to win:' pwin1 Say Return
pp: Procedure /*---------------------------------------------------------------------
- Compute and assign probabilities to get a sum x
- when throwing w dice each having s sides (marked from 1 to s)
- k=1 sets p1.*, k=2 sets p2.*
- --------------------------------------------------------------------*/
Use Arg k,w,s,p1.,p2. str= cnt.=0 Do wi=1 To w
str=str||'Do v'wi'=1 To' s';' End
str=str||'sum=' Do wi=1 To w-1
str=str||'v'wi'+' End
str=str||'v'w';' str=str||'cnt.'sum'+=1;' Do wi=1 To w
str=str||'End;' End
Interpret str psum=0 Do x=0 To w*s
If k=1 Then p1.x=cnt.x/(s**w) Else p2.x=cnt.x/(s**w) psum+=p1.x End
Return</lang>
- Output:
Player 1 has 9 dice with 4 sides each Player 2 has 6 dice with 6 sides each Probability for player 1 to win: 0.573144076782980082947530864198 Player 1 has 5 dice with 10 sides each Player 2 has 6 dice with 7 sides each Probability for player 1 to win: 0.642788628717626159168373721835
Algorithm using rational arithmetic
<lang oorexx>Numeric Digits 30 Call test '9 4 6 6' Call test '5 10 6 7' Exit
test: Parse Arg w1 s1 w2 s2 p1.='0/1' p2.='0/1' Call pp 1,w1,s1,p1.,p2. Call pp 2,w2,s2,p1.,p2. p2low.='0/1' Do x=w1 To w1*s1
Do y=0 To x-1 p2low.x=fr_add(p2low.x,p2.y) End End
pwin1='0/1' Do x=w1 To w1*s1
pwin1=fr_add(pwin1,fr_Mult(p1.x,p2low.x)) End
Say 'Player 1 has' w1 'dice with' s1 'sides each' Say 'Player 2 has' w2 'dice with' s2 'sides each' Say 'Probability for player 1 to win:' pwin1 Parse Var pwin1 nom '/' denom Say ' ->' (nom/denom) Say Return
pp: Procedure /*---------------------------------------------------------------------
- Compute and assign probabilities to get a sum x
- when throwing w dice each having s sides (marked from 1 to s)
- k=1 sets p1.*, k=2 sets p2.*
- --------------------------------------------------------------------*/
Use Arg k,w,s,p1.,p2. str= cnt.=0 Do wi=1 To w
str=str||'Do v'wi'=1 To' s';' End
str=str||'sum=' Do wi=1 To w-1
str=str||'v'wi'+' End
str=str||'v'w';' str=str||'cnt.sum+=1;' Do wi=1 To w
str=str||'End;' End
Interpret str
psum='0/1' Do x=0 To w*s
If k=1 Then p1.x=cnt.x'/'||(s**w) Else p2.x=cnt.x'/'||(s**w) psum=fr_add(psum,p1.x) End
Return
fr_add: Procedure
Parse Arg a,b
parse Var a an '/' az
parse Var b bn '/' bz
rn=an*bz+bn*az
rz=az*bz
res=fr_cancel(rn','rz)
Return res
fr_div: Procedure Parse Arg a,b parse Var a an '/' az parse Var b bn '/' bz rn=an*bz rz=az*bn res=fr_cancel(rn','rz) Return res
fr_mult: Procedure Parse Arg a,b parse Var a an '/' az parse Var b bn '/' bz rn=an*bn rz=az*bz res=fr_cancel(rn','rz) Return res
fr_cancel: Procedure Parse Arg n ',' z k=ggt(n,z) Return n%k'/'z%k
ggt: Procedure /**********************************************************************
- ggt (gcd) Greatest common Divisor
- Recursive procedure as shown in PL/I
- /
Parse Arg a,b if b = 0 then return abs(a) return ggt(b,a//b)</lang>
- Output:
Player 1 has 9 dice with 4 sides each
Player 2 has 6 dice with 6 sides each
Probability for player 1 to win: 48679795/84934656
-> 0.573144076782980082947530864198
Player 1 has 5 dice with 10 sides each
Player 2 has 6 dice with 7 sides each
Probability for player 1 to win: 3781171969/5882450000
-> 0.642788628717626159168373721834
Algorithm using class fraction
Class definition adapted from Arithmetic/Raional. <lang oorexx>Numeric Digits 50 Call test '9 4 6 6' Call test '5 10 6 7' Exit
test: Parse Arg w1 s1 w2 s2 p1.=.fraction~new(0,1) p2.=.fraction~new(0,1) Call pp 1,w1,s1,p1.,p2. Call pp 2,w2,s2,p1.,p2. p2low.=.fraction~new(0,1) Do x=w1 To w1*s1
Do y=0 To x-1 p2low.x=p2low.x+p2.y End End
pwin1=.fraction~new(0,1) Do x=w1 To w1*s1
pwin1=pwin1+(p1.x*p2low.x) End
Say 'Player 1 has' w1 'dice with' s1 'sides each' Say 'Player 2 has' w2 'dice with' s2 'sides each' Say 'Probability for player 1 to win:' pwin1~string Say ' ->' pwin1~tonumber Say Return
pp: Procedure /*---------------------------------------------------------------------
- Compute and assign probabilities to get a sum x
- when throwing w dice each having s sides (marked from 1 to s)
- k=1 sets p1.*, k=2 sets p2.*
- --------------------------------------------------------------------*/
Use Arg k,w,s,p1.,p2. str= cnt.=0 Do wi=1 To w
str=str||'Do v'wi'=1 To' s';' End
str=str||'sum=' Do wi=1 To w-1
str=str||'v'wi'+' End
str=str||'v'w';' str=str||'cnt.sum+=1;' Do wi=1 To w
str=str||'End;' End
Interpret str
psum=.fraction~new(0,1) Do x=0 To w*s
If k=1 Then p1.x=.fraction~new(cnt.x,s**w) Else p2.x=.fraction~new(cnt.x,s**w) psum=psum+p1.x End
Return
- class fraction inherit orderable
- options Digits 50
- method init
expose numerator denominator
use strict arg numerator, denominator = 1
--Trace ?R
--if numerator == 0 then denominator = 0
--else if denominator == 0 then raise syntax 98.900 array("Fraction denominator cannot be zero")
-- if the denominator is negative, make the numerator carry the sign
if denominator < 0 then do
numerator = -numerator
denominator = - denominator
end
-- find the greatest common denominator and reduce to -- the simplest form gcd = self~gcd(numerator~abs, denominator~abs)
numerator /= gcd denominator /= gcd
-- fraction instances are immutable, so these are -- read only attributes
-- calculate the greatest common denominator of a numerator/denominator pair
- method gcd private
use arg x, y
--Say 'gcd:' x y digits()
loop while y \= 0
-- check if they divide evenly
temp = x // y
x = y
y = temp
end
return x
-- calculate the least common multiple of a numerator/denominator pair
- method lcm private
use arg x, y return x / self~gcd(x, y) * y
- method abs
expose numerator denominator -- the denominator is always forced to be positive return self~class~new(numerator~abs, denominator)
-- convert a fraction to regular Rexx number
- method toNumber
expose numerator denominator if numerator == 0 then return 0 return numerator/denominator
- method add
expose numerator denominator use strict arg other -- convert to a fraction if a regular number if \other~isa(.fraction) then other = self~class~new(other, 1)
multiple = self~lcm(denominator, other~denominator) newa = numerator * multiple / denominator newb = other~numerator * multiple / other~denominator return self~class~new(newa + newb, multiple)
- method times
expose numerator denominator use strict arg other -- convert to a fraction if a regular number if \other~isa(.fraction) then other = self~class~new(other, 1) return self~class~new(numerator * other~numerator, denominator * other~denominator)
-- some operator overrides -- these only work if the left-hand-side of the -- subexpression is a quaternion
- method "*"
forward message("TIMES")
- method "+"
-- need to check if this is a prefix plus or an addition
if arg() == 0 then
return self -- we can return this copy since it is imutable
else
forward message("ADD")
- method string
expose numerator denominator if denominator == 1 then return numerator return numerator"/"denominator
-- override hashcode for collection class hash uses
- method hashCode
expose numerator denominator return numerator~hashcode~bitxor(numerator~hashcode)
- attribute numerator GET
- attribute denominator GET
- requires rxmath library</lang>
- Output:
Player 1 has 9 dice with 4 sides each
Player 2 has 6 dice with 6 sides each
Probability for player 1 to win: 48679795/84934656
-> 0.57314407678298008294753086419753086419753086419753
Player 1 has 5 dice with 10 sides each
Player 2 has 6 dice with 7 sides each
Probability for player 1 to win: 3781171969/5882450000
-> 0.64278862871762615916837372183358974576919480828566
Test
Result from 10 million tries. <lang oorexx>oid='diet.xxx'; Call sysFileDelete oid Call test '9 4 6 6' Call test '5 10 6 7' Exit test: Parse Arg n1 s1 n2 s2 Call o 'Player 1:' n1 'dice with' s1 'sides each' Call o 'Player 2:' n2 'dice with' s2 'sides each' cnt1.=0 cnt2.=0 win.=0 nn=10000000 Call time 'R' Do i=1 To nn
sum1=sum(n1 s1) ; cnt1.sum1+=1 sum2=sum(n2 s2) ; cnt2.sum2+=1 Select When sum1>sum2 Then win.1+=1 When sum1<sum2 Then win.2+=1 Otherwise win.0+=1 End End
Call o win.1/nn 'player 1 wins' Call o win.2/nn 'player 2 wins' Call o win.0/nn 'draws' /* Do i=min(n1,n2) To max(n1*s1,n2*s2)
Call o right(i,2) format(cnt1.i,7) format(cnt2.i,7) End
- /
Call o time('E') 'seconds elapsed' Return
sum: Parse Arg n s sum=0 Do k=1 To n
sum+=rand(s) End
Return sum
rand: Parse Arg n
Return random(n-1)+1
o: Say arg(1) Return lineout(oid,arg(1))</lang>
- Output:
Player 1: 9 dice with 4 sides each Player 2: 6 dice with 6 sides each 0.5730344 player 1 wins 0.3562464 player 2 wins 0.0707192 draws 186.794000 seconds elapsed Player 1: 5 dice with 10 sides each Player 2: 6 dice with 7 sides each 0.6425906 player 1 wins 0.312976 player 2 wins 0.0444334 draws 149.784000 seconds elapsed
Perl
<lang perl>use List::Util qw(sum0 max);
sub comb {
my ($s, $n) = @_;
$n || return (1);
my @r = (0) x ($n - max(@$s) + 1);
my @c = comb($s, $n - 1);
foreach my $i (0 .. $#c) {
$c[$i] || next;
foreach my $k (@$s) {
$r[$i + $k] += $c[$i];
}
}
return @r;
}
sub winning {
my ($s1, $n1, $s2, $n2) = @_;
my @p1 = comb($s1, $n1); my @p2 = comb($s2, $n2);
my ($win, $loss, $tie) = (0, 0, 0);
foreach my $i (0 .. $#p1) {
$win += $p1[$i] * sum0(@p2[0 .. $i - 1]);
$tie += $p1[$i] * sum0(@p2[$i .. $i ]);
$loss += $p1[$i] * sum0(@p2[$i+1 .. $#p2 ]);
}
my $total = sum0(@p1) * sum0(@p2);
map { $_ / $total } ($win, $tie, $loss);
}
print '(', join(', ', winning([1 .. 4], 9, [1 .. 6], 6)), ")\n"; print '(', join(', ', winning([1 .. 10], 5, [1 .. 7], 6)), ")\n";</lang>
- Output:
(0.57314407678298, 0.070766169838454, 0.356089753378566) (0.642788628717626, 0.0444960303104999, 0.312715340971874)
Phix
with javascript_semantics function throwDie(integer nSides, nDice, s, sequence counts) if nDice == 0 then counts[s] += 1 else for i=1 to nSides do counts = throwDie(nSides, nDice-1, s+i, counts) end for end if return counts end function function beatingProbability(integer nSides1, nDice1, nSides2, nDice2) integer len1 := (nSides1 + 1) * nDice1, len2 := (nSides2 + 1) * nDice2 sequence c1 = throwDie(nSides1, nDice1, 0, repeat(0,len1)), c2 = throwDie(nSides2, nDice2, 0, repeat(0,len2)) atom p12 := power(nSides1, nDice1) * power(nSides2, nDice2), tot := 0.0 for i=1 to len1 do for j=1 to min(i-1,len2) do tot += (c1[i] * c2[j]) / p12 end for end for return tot end function printf(1,"%0.16f\n",beatingProbability(4, 9, 6, 6)) printf(1,"%0.16f\n",beatingProbability(10, 5, 7, 6))
- Output:
(aside: the following tiny discrepancies are to be expected when using IEEE-754 64/80-bit floats; if you want to read anything into them, it should just be that we are all using the same hardware, and probably showing a couple too many digits of (in)accuracy on 32-bit.)
32 bit, same as Go, Kotlin
0.5731440767829815 0.6427886287176273
64 bit, same as D, Python[last], Ruby, Tcl
0.5731440767829801 0.6427886287176262
PL/I
version 1
<lang pli>*process source attributes xref;
dicegame: Proc Options(main);
Call test(9, 4,6,6);
Call test(5,10,6,7);
test: Proc(w1,s1,w2,s2);
Dcl (w1,s1,w2,s2,x,y) Bin Fixed(31);
Dcl p1(100) Dec Float(18) Init((100)0);
Dcl p2(100) Dec Float(18) Init((100)0);
Dcl p2low(100) Dec Float(18) Init((100)0);
Call pp(w1,s1,p1);
Call pp(w2,s2,p2);
Do x=w1 To w1*s1;
Do y=0 To x-1;
p2low(x)+=p2(y);
End;
End;
pwin1=0;
Do x=w1 To w1*s1;
pwin1+=p1(x)*p2low(x);
End;
Put Edit('Player 1 has ',w1,' dice with ',s1,' sides each')
(Skip,3(a,f(2)));
Put Edit('Player 2 has ',w2,' dice with ',s2,' sides each')
(Skip,3(a,f(2)));
Put Edit('Probability for player 1 to win: ',pwin1)(Skip,a,f(20,18));
Put Edit()(Skip,a);
End;
pp: Proc(w,s,p);
/*--------------------------------------------------------------------
* Compute and assign probabilities to get a sum x
* when throwing w dice each having s sides (marked from 1 to s)
*-------------------------------------------------------------------*/
Dcl (w,s) Bin Fixed(31);
Dcl p(100) Dec Float(18);
Dcl cnt(100) Bin Fixed(31);
Dcl (a(12),e(12),v(12),sum,i,n) Bin Fixed(31);
a=0;
e=0;
Do i=1 To w;
a(i)=1;
e(i)=s;
End;
n=0;
cnt=0;
Do v(1)=a(1) To e(1);
Do v(2)=a(2) To e(2);
Do v(3)=a(3) To e(3);
Do v(4)=a(4) To e(4);
Do v(5)=a(5) To e(5);
Do v(6)=a(6) To e(6);
Do v(7)=a(7) To e(7);
Do v(8)=a(8) To e(8);
Do v(9)=a(9) To e(9);
Do v(10)=a(10) To e(10);
sum=0;
Do i=1 To 10;
sum=sum+v(i);
End;
cnt(sum)+=1;
n+=1;
End;
End;
End;
End;
End;
End;
End;
End;
End;
End;
Do k=w To w*s;
p(k)=divide(cnt(k),n,18,16);
End;
End;
End;</lang>
- Output:
Player 1 has 9 dice with 4 sides each Player 2 has 6 dice with 6 sides each Probability for player 1 to win: 0.573013663291931152 Player 1 has 5 dice with 10 sides each Player 2 has 6 dice with 7 sides each Probability for player 1 to win: 0.642703175544738770
version 2 using rational arithmetic
<lang pli>*process source attributes xref;
dgf: Proc Options(main);
Call test(9, 4,6,6);
Call test(5,10,6,7);
test: Proc(w1,s1,w2,s2);
Dcl (w1,s1,w2,s2,x,y) Dec Float(18);
Dcl 1 p1(100),
2 nom Dec Float(18) Init((100)0),
2 denom Dec Float(18) Init((100)1);
Dcl 1 p2(100),
2 nom Dec Float(18) Init((100)0),
2 denom Dec Float(18) Init((100)1);
Dcl 1 p2low(100),
2 nom Dec Float(18) Init((100)0),
2 denom Dec Float(18) Init((100)1);
Dcl 1 pwin1,
2 nom Dec Float(18) Init(0),
2 denom Dec Float(18) Init(1);
Dcl 1 prod Like pwin1;
Call pp(w1,s1,p1);
Call pp(w2,s2,p2);
Do x=w1 To w1*s1;
Do y=0 To x-1;
Call fr_add(p2low(x),p2(y),p2low(x));
End;
End;
Do x=w1 To w1*s1;
Call fr_mult(p1(x),p2low(x),prod);
Call fr_add(pwin1,prod,pwin1);
End;
Put Edit('Player 1 has ',w1,' dice with ',s1,' sides each')
(Skip,3(a,f(2)));
Put Edit('Player 2 has ',w2,' dice with ',s2,' sides each')
(Skip,3(a,f(2)));
Put Edit('Probability for player 1 to win: ',
str(pwin1.nom),'/',str(pwin1.denom))(Skip,4(a));
Put Edit(' -> ',
pwin1.nom/pwin1.denom)(Skip,a,f(20,18));
Put Edit()(Skip,a);
End;
pp: Proc(w,s,p);
/*--------------------------------------------------------------------
* Compute and assign probabilities to get a sum x
* when throwing w dice each having s sides (marked from 1 to s)
*-------------------------------------------------------------------*/
Dcl (w,s) Dec Float(18);
Dcl 1 p(100),
2 nom Dec Float(18),
2 denom Dec Float(18);
Dcl cnt(100) Dec Float(18);
Dcl (a(12),e(12),v(12),sum,i,n) Dec Float(18);
a=0;
e=0;
Do i=1 To w;
a(i)=1;
e(i)=s;
End;
n=0;
cnt=0;
Do v(1)=a(1) To e(1);
Do v(2)=a(2) To e(2);
Do v(3)=a(3) To e(3);
Do v(4)=a(4) To e(4);
Do v(5)=a(5) To e(5);
Do v(6)=a(6) To e(6);
Do v(7)=a(7) To e(7);
Do v(8)=a(8) To e(8);
Do v(9)=a(9) To e(9);
Do v(10)=a(10) To e(10);
sum=0;
Do i=1 To 10;
sum=sum+v(i);
End;
cnt(sum)+=1;
n+=1;
End;
End;
End;
End;
End;
End;
End;
End;
End;
End;
Do k=w To w*s;
p(k).nom=cnt(k);
p(k).denom=n;
End;
End;
fr_add: Proc(a,b,res);
Dcl 1 a,
2 nom Dec Float(18),
2 denom Dec Float(18);
Dcl 1 b Like a;
Dcl res like a;
/* Put Edit('fr_add',a.nom,a.denom,b.nom,b.denom)(Skip,a,4(f(15))); */
res.nom=a.nom*b.denom+b.nom*a.denom;
res.denom=a.denom*b.denom;
Call fr_cancel(res,res);
End;
fr_mult: Proc(a,b,res);
Dcl 1 a,
2 nom Dec Float(18),
2 denom Dec Float(18);
Dcl 1 b Like a;
Dcl res like a;
/* Put Edit('fr_mult',a.nom,a.denom,b.nom,b.denom)(Skip,a,4(f(15)));*/
res.nom=a.nom*b.nom;
res.denom=a.denom*b.denom;
Call fr_cancel(res,res);
End;
fr_cancel: Proc(a,res);
Dcl 1 a,
2 nom Dec Float(18),
2 denom Dec Float(18);
Dcl k Dec Float(18);
Dcl 1 res like a;
/* Put Edit('fr_cancel',a.nom,a.denom)(Skip,a,4(f(15))); */
k=ggt(a.nom,a.denom);
res=a/k;
End;
ggt: Proc(a,b) Recursive Returns(Dec Float(18)); /********************************************************************** * ggt (gcd) Greatest common Divisor * Recursive Proc(a,b)) as shown in PL/I **********************************************************************/ Dcl (a,b) Dec Float(18); if b = 0 then return (abs(a)); return (ggt(b,mod(a,b))); End;
str: Proc(x) Returns(Char(20) Var); Dcl x Dec Float(18); Dcl res Char(20) Var; Put String(res) Edit(x)(f(20)); Return (trim(res)); End;
End;</lang>
- Output:
Player 1 has 9 dice with 4 sides each
Player 2 has 6 dice with 6 sides each
Probability for player 1 to win: 48679795/84934656
-> 0.573144076782980083
Player 1 has 5 dice with 10 sides each
Player 2 has 6 dice with 7 sides each
Probability for player 1 to win: 3781171969/5882450000
-> 0.642788628717626159
Python
<lang python>from itertools import product
def gen_dict(n_faces, n_dice):
counts = [0] * ((n_faces + 1) * n_dice)
for t in product(range(1, n_faces + 1), repeat=n_dice):
counts[sum(t)] += 1
return counts, n_faces ** n_dice
def beating_probability(n_sides1, n_dice1, n_sides2, n_dice2):
c1, p1 = gen_dict(n_sides1, n_dice1) c2, p2 = gen_dict(n_sides2, n_dice2) p12 = float(p1 * p2)
return sum(p[1] * q[1] / p12
for p, q in product(enumerate(c1), enumerate(c2))
if p[0] > q[0])
print beating_probability(4, 9, 6, 6) print beating_probability(10, 5, 7, 6)</lang>
- Output:
0.573144076783 0.642788628718
To handle larger number of dice (and faster in general): <lang python>from __future__ import print_function, division
def combos(sides, n):
if not n: return [1]
ret = [0] * (max(sides)*n + 1)
for i,v in enumerate(combos(sides, n - 1)):
if not v: continue
for s in sides: ret[i + s] += v
return ret
def winning(sides1, n1, sides2, n2):
p1, p2 = combos(sides1, n1), combos(sides2, n2)
win,loss,tie = 0,0,0 # 'win' is 1 beating 2
for i,x1 in enumerate(p1):
# using accumulated sum on p2 could save some time
win += x1*sum(p2[:i])
tie += x1*sum(p2[i:i+1])
loss+= x1*sum(p2[i+1:])
s = sum(p1)*sum(p2)
return win/s, tie/s, loss/s
print(winning(range(1,5), 9, range(1,7), 6)) print(winning(range(1,11), 5, range(1,8), 6)) # this seem hardly fair
- mountains of dice test case
- print(winning((1, 2, 3, 5, 9), 700, (1, 2, 3, 4, 5, 6), 800))</lang>
- Output:
(0.5731440767829801, 0.070766169838454, 0.3560897533785659) (0.6427886287176262, 0.044496030310499875, 0.312715340971874)
If we further restrict die faces to be 1 to n instead of arbitrary values, the combo generation can be made much faster: <lang python>from __future__ import division, print_function from itertools import accumulate # Python3 only
def combos(sides, n):
ret = [1] + [0]*(n + 1)*sides # extra length for negative indices
for p in range(1, n + 1):
rolling_sum = 0
for i in range(p*sides, p - 1, -1):
rolling_sum += ret[i - sides] - ret[i]
ret[i] = rolling_sum
ret[p - 1] = 0
return ret
def winning(d1, n1, d2, n2):
c1, c2 = combos(d1, n1), combos(d2, n2) ac = list(accumulate(c2 + [0]*(len(c1) - len(c2))))
return sum(v*a for v,a in zip(c1[1:], ac)) / (ac[-1]*sum(c1))
print(winning(4, 9, 6, 6))
print(winning(5, 10, 6, 7))
- print(winning(6, 700, 8, 540))</lang>
- Output:
0.5731440767829801 0.6427886287176262
R
Solving these sorts of problems by simulation is trivial in R. <lang r>probability <- function(facesCount1, diceCount1, facesCount2, diceCount2) {
mean(replicate(10^6, sum(sample(facesCount1, diceCount1, replace = TRUE)) > sum(sample(facesCount2, diceCount2, replace = TRUE))))
} cat("Player 1's probability of victory is", probability(4, 9, 6, 6),
"in the first game and", probability(10, 5, 7, 6), "in the second.")</lang>
- Output:
Player 1's probability of victory is 0.572652 in the first game and 0.642817 in the second.
Racket
<lang racket>#lang racket
(define probs# (make-hash))
(define (NdD n d)
(hash-ref!
probs# (cons n d)
(λ ()
(cond
[(= n 0) ; every chance of nothing!
(hash 0 1)]
[else
(for*/fold ((hsh (hash))) (((i p) (in-hash (NdD (sub1 n) d))) (r (in-range 1 (+ d 1))))
(hash-update hsh (+ r i) (λ (p+) (+ p+ (/ p d))) 0))]))))
(define (game-probs N1 D1 N2 D2)
(define P1 (NdD N1 D1))
(define P2 (NdD N2 D2))
(define-values (W D L)
(for*/fold ((win 0) (draw 0) (lose 0)) (((r1 p1) (in-hash P1)) ((r2 p2) (in-hash P2)))
(define p (* p1 p2))
(cond
[(< r1 r2) (values win draw (+ lose p))]
[(= r1 r2) (values win (+ draw p) lose)]
[(> r1 r2) (values (+ win p) draw lose)])))
(printf "P(P1 win): ~a~%" (real->decimal-string W 6))
(printf "P(draw): ~a~%" (real->decimal-string D 6))
(printf "P(P2 win): ~a~%" (real->decimal-string L 6))
(list W D L))
(printf "GAME 1 (9D4 vs 6D6)~%") (game-probs 9 4 6 6) (newline)
(printf "GAME 2 (5D10 vs 6D7) [what is a D7?]~%") (game-probs 5 10 6 7)</lang>
- Output:
GAME 1 (9D4 vs 6D6) P(P1 win): 0.573144 P(draw): 0.070766 P(P2 win): 0.356090 (48679795/84934656 144252007/2038431744 725864657/2038431744) GAME 2 (5D10 vs 6D7) [what is a D7?] P(P1 win): 0.642789 P(draw): 0.044496 P(P2 win): 0.312715 (3781171969/5882450000 523491347/11764900000 735812943/2352980000)
Raku
(formerly Perl 6)
<lang perl6>sub likelihoods ($roll) {
my ($dice, $faces) = $roll.comb(/\d+/); my @counts; @counts[$_]++ for [X+] |(1..$faces,) xx $dice; return [@counts[]:p], $faces ** $dice;
}
sub beating-probability ([$roll1, $roll2]) {
my (@c1, $p1) := likelihoods $roll1;
my (@c2, $p2) := likelihoods $roll2;
my $p12 = $p1 * $p2;
[+] gather for flat @c1 X @c2 -> $p, $q {
take $p.value * $q.value / $p12 if $p.key > $q.key;
}
}
- We're using standard DnD notation for dice rolls here.
say .gist, "\t", .raku given beating-probability < 9d4 6d6 >; say .gist, "\t", .raku given beating-probability < 5d10 6d7 >;</lang>
- Output:
0.573144077 <48679795/84934656> 0.64278862872 <3781171969/5882450000>
Note that all calculations are in integer and rational arithmetic, so the results in fractional notation are exact.
REXX
version 1
(adapted for Classic Rexx) <lang rexx>/* REXX */ Numeric Digits 30 Call test '9 4 6 6' Call test '5 10 6 7' Exit test: Parse Arg w1 s1 w2 s2 plist1=pp(w1,s1) p1.=0 Do x=w1 To w1*s1
Parse Var plist1 p1.x plist1 End
plist2=pp(w2,s2) p2.=0 Do x=w2 To w2*s2
Parse Var plist2 p2.x plist2 End
p2low.=0 Do x=w1 To w1*s1
Do y=0 To x-1 p2low.x=p2low.x+p2.y End End
pwin1=0 Do x=w1 To w1*s1
pwin1=pwin1+p1.x*p2low.x End
Say 'Player 1 has' w1 'dice with' s1 'sides each' Say 'Player 2 has' w2 'dice with' s2 'sides each' Say 'Probability for player 1 to win:' pwin1 Say Return
pp: Procedure /*---------------------------------------------------------------------
- Compute and return the probabilities to get a sum x
- when throwing w dice each having s sides (marked from 1 to s)
- --------------------------------------------------------------------*/
Parse Arg w,s str= cnt.=0 Do wi=1 To w
str=str||'Do v'wi'=1 To' s';' End
str=str||'sum=' Do wi=1 To w-1
str=str||'v'wi'+' End
str=str||'v'w';' str=str||'cnt.'sum'=cnt.'sum'+1;' Do wi=1 To w
str=str||'End;' End
Interpret str psum=0 Do x=0 To w*s
p.x=cnt.x/(s**w) psum=psum+p.x End
res= Do x=w To s*w
res=res p.x End
Return res</lang>
- Output:
Player 1 has 9 dice with 4 sides each Player 2 has 6 dice with 6 sides each Probability for player 1 to win: 0.573144076782980082947530864198 Player 1 has 5 dice with 10 sides each Player 2 has 6 dice with 7 sides each Probability for player 1 to win: 0.642788628717626159168373721835
version 2
<lang rexx>/* REXX */ oid='diet.xxx'; 'erase' oid Call test '9 4 6 6' Call test '5 10 6 7' Exit test: Parse Arg n1 s1 n2 s2 Call o 'Player 1:' n1 'dice with' s1 'sides each' Call o 'Player 2:' n2 'dice with' s2 'sides each' cnt1.=0 cnt2.=0 win.=0 nn=10000 Call time 'R' Do i=1 To nn
sum1=sum(n1 s1) ; cnt1.sum1=cnt1.sum1+1 sum2=sum(n2 s2) ; cnt2.sum2=cnt2.sum2+1 Select When sum1>sum2 Then win.1=win.1+1 When sum1<sum2 Then win.2=win.2+1 Otherwise win.0=win.0+1 End End
Call o win.1/nn 'player 1 wins' Call o win.2/nn 'player 2 wins' Call o win.0/nn 'draws' /* Do i=min(n1,n2) To max(n1*s1,n2*s2)
Call o right(i,2) format(cnt1.i,7) format(cnt2.i,7) End
- /
Call o time('E') 'seconds elapsed' Return
sum: Parse Arg n s sum=0 Do k=1 To n
sum=sum+rand(s) End
Return sum
rand: Parse Arg n
Return random(n-1)+1
o: Say arg(1) Return lineout(oid,arg(1))</lang>
- Output:
Player 1: 9 dice with 4 sides each Player 2: 6 dice with 6 sides each 0.574 player 1 wins 0.3506 player 2 wins 0.0754 draws 0.109000 seconds elapsed Player 1: 5 dice with 10 sides each Player 2: 6 dice with 7 sides each 0.6411 player 1 wins 0.3115 player 2 wins 0.0474 draws 0.078000 seconds elapsed
optimized
This REXX version is an optimized and reduced version of the first part of the 1st REXX example. <lang rexx>/*REXX pgm computes and displays the probabilities of a two─player S─sided, N─dice game.*/ numeric digits 100 /*increase/decrease to heart's desire. */ call game 9 4, 6 6 /*1st player: 9 dice, 4 sides; 2nd player: 6 dice, 6 sides*/ call game 5 10, 6 7 /* " " 5 " 10 " " " 6 " 7 " */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ game: parse arg w.1 s.1, w.2 s.2 /*1st player(dice sides), 2nd player···*/
p.= 0
do j=1 for 2; @@.j= prob(w.j, s.j)
do k=w.j to w.j*s.j; parse var @@.j p.j.k @@.j; end /*k*/
end /*j*/
low.= 0
do j=w.1 to w.1*s.1
do k=0 for j; low.j= low.j + p.2.k; end /*k*/
end /*j*/
say ' Player 1 has ' w.1 " dice with " s.1 ' sides each.'
say ' Player 2 has ' w.2 " dice with " s.2 ' sides each.'
winP= 0
do j=w.1 to w.1*s.1; winP= winP + p.1.j * low.j
end /*j*/
say 'The probability for first player to win is ' format(winP*100,,digits()%2) "%."
say /* ↑ */
return /*show 1/2 of 100 dec. digits────┘ */
/*──────────────────────────────────────────────────────────────────────────────────────*/ prob: procedure; parse arg n,s,,@ $; #.= 0; pow= s**n
do j=1 for n; @= @'DO _'j"=1 FOR" s';'; end /*j*/
@= @'_='; do k=1 for n-1; @= @"_"k'+' ; end /*k*/
interpret @'_'n";#."_'=#.'_"+1"copies(';END', k)
ns= n*s; do j=0 to ns; p.j= #.j / pow; end /*j*/
do k=n to ns; $= $ p.k; end /*k*/
return $ /* ◄──────────────── probability of 1st player to win, S─sided, N dice.*/</lang>
- output when using the default inputs:
Player 1 has 9 dice with 4 sides each. Player 2 has 6 dice with 6 sides each. The probability for first player to win is 57.31440767829800829475308641975308641975308641975309 %. Player 1 has 5 dice with 10 sides each. Player 2 has 6 dice with 7 sides each. The probability for first player to win is 64.27886287176261591683737218335897457691948082856633 %.
Ruby
<lang ruby>def roll_dice(n_dice, n_faces)
return 0,1 if n_dice.zero? one = [1] * n_faces zero = [0] * (n_faces-1) (1...n_dice).inject(one){|ary,_| (zero + ary + zero).each_cons(n_faces).map{|a| a.inject(:+)} }.map.with_index(n_dice){|n,sum| [sum,n]} # sum: total of the faces
end
def game(dice1, faces1, dice2, faces2)
p1 = roll_dice(dice1, faces1) p2 = roll_dice(dice2, faces2) p1.product(p2).each_with_object([0,0,0]) do |((sum1, n1), (sum2, n2)), win| win[sum1 <=> sum2] += n1 * n2 # [0]:draw, [1]:win, [-1]:lose end
end
[[9, 4, 6, 6], [5, 10, 6, 7]].each do |d1, f1, d2, f2|
puts "player 1 has #{d1} dice with #{f1} faces each"
puts "player 2 has #{d2} dice with #{f2} faces each"
win = game(d1, f1, d2, f2)
sum = win.inject(:+)
puts "Probability for player 1 to win: #{win[1]} / #{sum}",
" -> #{win[1].fdiv(sum)}", ""
end</lang>
- Output:
player 1 has 9 dice with 4 faces each
player 2 has 6 dice with 6 faces each
Probability for player 1 to win: 7009890480 / 12230590464
-> 0.5731440767829801
player 1 has 5 dice with 10 faces each
player 2 has 6 dice with 7 faces each
Probability for player 1 to win: 7562343938 / 11764900000
-> 0.6427886287176262
Rust
<lang rust>// Returns a vector containing the number of ways each possible sum of face // values can occur. fn get_totals(dice: usize, faces: usize) -> Vec<f64> {
let mut result = vec![1.0; faces + 1];
for d in 2..=dice {
let mut tmp = vec![0.0; d * faces + 1];
for i in d - 1..result.len() {
for j in 1..=faces {
tmp[i + j] += result[i];
}
}
result = tmp;
}
result
}
fn probability(dice1: usize, faces1: usize, dice2: usize, faces2: usize) -> f64 {
let totals1 = get_totals(dice1, faces1);
let totals2 = get_totals(dice2, faces2);
let mut wins = 0.0;
let mut total = 0.0;
for i in dice1..totals1.len() {
for j in dice2..totals2.len() {
let n = totals1[i] * totals2[j];
total += n;
if j < i {
wins += n;
}
}
}
wins / total
}
fn main() {
println!("{}", probability(9, 4, 6, 6));
println!("{}", probability(5, 10, 6, 7));
}</lang>
- Output:
0.5731440767829801 0.6427886287176262
Sidef
<lang ruby>func combos(sides, n) {
n || return [1]
var ret = ([0] * (n*sides.max + 1))
combos(sides, n-1).each_kv { |i,v|
v && for s in sides { ret[i + s] += v }
}
return ret
}
func winning(sides1, n1, sides2, n2) {
var (p1, p2) = (combos(sides1, n1), combos(sides2, n2))
var (win,loss,tie) = (0,0,0)
p1.each_kv { |i, x|
win += x*p2.ft(0,i-1).sum
tie += x*p2.ft(i, i).sum
loss += x*p2.ft(i+1).sum
}
[win, tie, loss] »/» p1.sum*p2.sum
}
func display_results(String title, Array res) {
say "=> #{title}"
for name, prob in (%w(p₁\ win tie p₂\ win) ~Z res) {
say "P(#{'%6s' % name}) =~ #{prob.round(-11)} (#{prob.as_frac})"
}
print "\n"
}
display_results('9D4 vs 6D6', winning(range(1, 4), 9, range(1,6), 6)) display_results('5D10 vs 6D7', winning(range(1,10), 5, range(1,7), 6))</lang>
- Output:
=> 9D4 vs 6D6 P(p₁ win) =~ 0.57314407678 (48679795/84934656) P( tie) =~ 0.07076616984 (144252007/2038431744) P(p₂ win) =~ 0.35608975338 (725864657/2038431744) => 5D10 vs 6D7 P(p₁ win) =~ 0.64278862872 (3781171969/5882450000) P( tie) =~ 0.04449603031 (523491347/11764900000) P(p₂ win) =~ 0.31271534097 (735812943/2352980000)
Tcl
To handle the nested loop in NdK, Tcl's metaprogramming abilities are exercised. The goal is to produce a script that looks like: <lang Tcl>foreach d0 {1 2 3 4 5 6} {
foreach d1 {1 2 3 4 5 6} {
...
foreach dN {1 2 3 4 5 6} {
dict incr sum [::tcl::mathop::+ $n $d0 $d1 ... $DN]
}
...
}
}</lang>
See the comments attached to that procedure for a more thorough understanding of how that is achieved (with the caveat that $d0..$dN are reversed).
Such metaprogramming is a very powerful technique in Tcl for building scripts where other approaches (in this case, recursion) might not be appealing, and should be in every programmer's toolbox!
<lang Tcl>proc range {b} { ;# a common standard proc: [range 5] -> {0 1 2 3 4}
set a 0
set res {}
while {$a < $b} {
lappend res $a
incr a
}
return $res
}
- This proc builds up a nested foreach call, then evaluates it.
- The script is built up in $script, starting with the body using "%%" as
- a placeholder.
- For each die, a level is wrapped around it as follows:
- set script {foreach d0 {1 2 3 4 5 6} $script}
- set script {foreach d1 {1 2 3 4 5 6} $script}
- .. and {$d0 $d1 ..} are collected in the variable $vars, which is used
- to replace "%%" at the end.
- The script is evaluated with [try] - earlier Tcl's could use [catch] or [eval]
proc NdK {n {k 6}} { ;# calculate a score histogram for $n dice of $k faces
set sum {}
set script {
dict incr sum [::tcl::mathop::+ $n %%] ;# add $n because ranges are 0-based
} ;# %% is a placeholder
set vars ""
for {set i 0} {$i < $n} {incr i} {
set script [list foreach d$i [range $k] $script]
append vars " \$d$i"
}
set script [string map [list %% $vars] $script]
try $script
return $sum
}
proc win_pr {p1 p2} { ;# calculate the winning probability of player 1 given two score histograms
set P 0
set N 0
dict for {d1 k1} $p1 {
dict for {d2 k2} $p2 {
set k [expr {$k1 * $k2}]
incr N $k
incr P [expr {$k * ($d1 > $d2)}]
}
}
expr {$P * 1.0 / $N}
}
foreach {p1 p2} {
{9 4} {6 6}
{5 10} {6 7}
} {
puts [format "p1 has %dd%d; p2 has %dd%d" {*}$p1 {*}$p2]
puts [format " p1 wins with Pr(%s)" [win_pr [NdK {*}$p1] [NdK {*}$p2]]]
}</lang>
- Output:
p1 has 9d4; p2 has 6d6 p1 wins with Pr(0.5731440767829801) p1 has 5d10; p2 has 6d7 p1 wins with Pr(0.6427886287176262)
Factoring out foreach*
The nested-loop generation illustrated above is useful to factor out as a routine by itself. Here it is abstracted as foreach*, with NdK modified to suit.
I include this to emphasise the importance and power of metaprogramming in a Tcler's toolbox, as well as sharing a useful proc.
<lang Tcl>package require Tcl 8.6 ;# for [tailcall] - otherwise use [uplevel 1 $script]
- This proc builds up a nested foreach call, then evaluates it.
- this:
- foreach* a {1 2 3} b {4 5 6} {
- puts "$a + $b"
- }
- becomes:
- foreach a {1 2 3} {
- foreach b {4 5 6} {
- puts "$a + $b"
- }
- }
proc foreach* {args} {
set script [lindex $args end]
set args [lrange $args 0 end-1]
foreach {b a} [lreverse $args] {
set script [list foreach $a $b $script]
}
tailcall {*}$script
}
proc NdK {n {k 6}} { ;# calculate a score histogram for $n dice of $k faces
set args {} ;# arguments to [foreach*]
set vars {} ;# variables used in [foreach*] arguments that need to be added to sum
set sum {} ;# this will be the result dictionary
for {set i 0} {$i < $n} {incr i} {
lappend args d$i [range $k]
lappend vars "\$d$i"
}
set vars [join $vars +]
# [string map] to avoid "Quoting Hell"
set script [string map [list %% $vars] {
dict incr sum [expr {$n + %%}] ;# $n because [range] is 0-based
}]
foreach* {*}$args $script
return $sum
}</lang>
Wren
<lang ecmascript>var throwDie // recursive throwDie = Fn.new { |nSides, nDice, s, counts|
if (nDice == 0) {
counts[s] = counts[s] + 1
return
}
for (i in 1..nSides) throwDie.call(nSides, nDice-1, s + i, counts)
}
var beatingProbability = Fn.new { |nSides1, nDice1, nSides2, nDice2|
var len1 = (nSides1 + 1) * nDice1 var c1 = List.filled(len1, 0) throwDie.call(nSides1, nDice1, 0, c1)
var len2 = (nSides2 + 1) * nDice2 var c2 = List.filled(len2, 0) throwDie.call(nSides2, nDice2, 0, c2)
var p12 = nSides1.pow(nDice1) * nSides2.pow(nDice2)
var tot = 0
for (i in 0...len1) {
for (j in 0...i.min(len2)) {
tot = tot + c1[i] * c2[j] / p12
}
}
return tot
}
System.print(beatingProbability.call(4, 9, 6, 6)) System.print(beatingProbability.call(10, 5, 7, 6))</lang>
- Output:
0.57314407678298 0.64278862871763
zkl
<lang zkl>fcn combos(sides, n){
if(not n) return(T(1));
ret:=((0).max(sides)*n + 1).pump(List(),0);
foreach i,v in (combos(sides, n - 1).enumerate()){
if(not v) continue;
foreach s in (sides){ ret[i + s] += v }
}
ret
}
fcn winning(sides1,n1, sides2,n2){
p1, p2 := combos(sides1, n1), combos(sides2, n2);
win,loss,tie := 0,0,0; # 'win' is 1 beating 2
foreach i,x1 in (p1.enumerate()){
# using accumulated sum on p2 could save some time
win += x1*p2[0,i].sum(0);
tie += x1*p2[i,1].sum(0); // i>p2.len() but p2[bigi,?]-->[]
loss+= x1*p2[i+1,*].sum(0);
}
s := p1.sum(0)*p2.sum(0);
return(win.toFloat()/s, tie.toFloat()/s, loss.toFloat()/s);
}</lang> <lang zkl>println(winning([1..4].walk(), 9, [1..6].walk(),6)); println(winning([1..10].walk(),5, [1..7].walk(),6)); # this seem hardly fair</lang>
- Output:
L(0.573144,0.0707662,0.35609) L(0.642789,0.044496,0.312715)