Dice game probabilities: Difference between revisions
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=={{header|PL/I}}== |
=={{header|PL/I}}== |
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===version 1=== |
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<lang pli>*process source attributes xref; |
<lang pli>*process source attributes xref; |
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dicegame: Proc Options(main); |
dicegame: Proc Options(main); |
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Line 511: | Line 512: | ||
Player 2 has 6 dice with 7 sides each |
Player 2 has 6 dice with 7 sides each |
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Probability for player 1 to win: 0.642703175544738770</pre> |
Probability for player 1 to win: 0.642703175544738770</pre> |
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===version 1 (fraction arithmetic)=== |
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<lang pli>*process source attributes xref; |
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dgf: Proc Options(main); |
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Call test(9, 4,6,6); |
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Call test(5,10,6,7); |
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test: Proc(w1,s1,w2,s2); |
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Dcl (w1,s1,w2,s2,x,y) Dec Float(18); |
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Dcl 1 p1(100), |
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2 nom Dec Float(18) Init((100)0), |
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2 denom Dec Float(18) Init((100)1); |
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Dcl 1 p2(100), |
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2 nom Dec Float(18) Init((100)0), |
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2 denom Dec Float(18) Init((100)1); |
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Dcl 1 p2low(100), |
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2 nom Dec Float(18) Init((100)0), |
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2 denom Dec Float(18) Init((100)1); |
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Dcl 1 pwin1, |
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2 nom Dec Float(18) Init(0), |
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2 denom Dec Float(18) Init(1); |
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Dcl 1 prod Like pwin1; |
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Call pp(w1,s1,p1); |
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Call pp(w2,s2,p2); |
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Do x=w1 To w1*s1; |
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Do y=0 To x-1; |
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Call fr_add(p2low(x),p2(y),p2low(x)); |
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End; |
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End; |
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Do x=w1 To w1*s1; |
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Call fr_mult(p1(x),p2low(x),prod); |
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Call fr_add(pwin1,prod,pwin1); |
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End; |
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Put Edit('Player 1 has ',w1,' dice with ',s1,' sides each') |
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(Skip,3(a,f(2))); |
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Put Edit('Player 2 has ',w2,' dice with ',s2,' sides each') |
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(Skip,3(a,f(2))); |
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Put Edit('Probability for player 1 to win: ', |
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str(pwin1.nom),'/',str(pwin1.denom))(Skip,4(a)); |
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Put Edit(' -> ', |
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pwin1.nom/pwin1.denom)(Skip,a,f(20,18)); |
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Put Edit('')(Skip,a); |
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End; |
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pp: Proc(w,s,p); |
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/*-------------------------------------------------------------------- |
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* Compute and assign probabilities to get a sum x |
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* when throwing w dice each having s sides (marked from 1 to s) |
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*-------------------------------------------------------------------*/ |
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Dcl (w,s) Dec Float(18); |
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Dcl 1 p(100), |
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2 nom Dec Float(18), |
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2 denom Dec Float(18); |
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Dcl cnt(100) Dec Float(18); |
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Dcl (a(12),e(12),v(12),sum,i,n) Dec Float(18); |
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a=0; |
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e=0; |
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Do i=1 To w; |
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a(i)=1; |
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e(i)=s; |
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End; |
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n=0; |
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cnt=0; |
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Do v(1)=a(1) To e(1); |
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Do v(2)=a(2) To e(2); |
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Do v(3)=a(3) To e(3); |
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Do v(4)=a(4) To e(4); |
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Do v(5)=a(5) To e(5); |
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Do v(6)=a(6) To e(6); |
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Do v(7)=a(7) To e(7); |
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Do v(8)=a(8) To e(8); |
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Do v(9)=a(9) To e(9); |
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Do v(10)=a(10) To e(10); |
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sum=0; |
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Do i=1 To 10; |
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sum=sum+v(i); |
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End; |
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cnt(sum)+=1; |
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n+=1; |
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End; |
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End; |
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End; |
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End; |
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End; |
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End; |
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End; |
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End; |
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End; |
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End; |
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Do k=w To w*s; |
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p(k).nom=cnt(k); |
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p(k).denom=n; |
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End; |
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End; |
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fr_add: Proc(a,b,res); |
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Dcl 1 a, |
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2 nom Dec Float(18), |
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2 denom Dec Float(18); |
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Dcl 1 b Like a; |
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Dcl res like a; |
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/* Put Edit('fr_add',a.nom,a.denom,b.nom,b.denom)(Skip,a,4(f(15))); */ |
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res.nom=a.nom*b.denom+b.nom*a.denom; |
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res.denom=a.denom*b.denom; |
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Call fr_cancel(res,res); |
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End; |
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fr_mult: Proc(a,b,res); |
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Dcl 1 a, |
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2 nom Dec Float(18), |
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2 denom Dec Float(18); |
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Dcl 1 b Like a; |
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Dcl res like a; |
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/* Put Edit('fr_mult',a.nom,a.denom,b.nom,b.denom)(Skip,a,4(f(15)));*/ |
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res.nom=a.nom*b.nom; |
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res.denom=a.denom*b.denom; |
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Call fr_cancel(res,res); |
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End; |
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fr_cancel: Proc(a,res); |
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Dcl 1 a, |
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2 nom Dec Float(18), |
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2 denom Dec Float(18); |
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Dcl k Dec Float(18); |
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Dcl 1 res like a; |
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/* Put Edit('fr_cancel',a.nom,a.denom)(Skip,a,4(f(15))); */ |
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k=ggt(a.nom,a.denom); |
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res=a/k; |
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End; |
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ggt: Proc(a,b) Recursive Returns(Dec Float(18)); |
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/********************************************************************** |
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* ggt (gcd) Greatest common Divisor |
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* Recursive Proc(a,b)) as shown in PL/I |
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**********************************************************************/ |
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Dcl (a,b) Dec Float(18); |
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if b = 0 then return (abs(a)); |
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return (ggt(b,mod(a,b))); |
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End; |
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str: Proc(x) Returns(Char(20) Var); |
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Dcl x Dec Float(18); |
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Dcl res Char(20) Var; |
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Put String(res) Edit(x)(f(20)); |
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Return (trim(res)); |
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End; |
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End;</lang> |
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{{out}} |
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<pre>Player 1 has 9 dice with 4 sides each |
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Player 2 has 6 dice with 6 sides each |
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Probability for player 1 to win: 48679795/84934656 |
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-> 0.573144076782980083 |
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Player 1 has 5 dice with 10 sides each |
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Player 2 has 6 dice with 7 sides each |
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Probability for player 1 to win: 3781171969/5882450000 |
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-> 0.642788628717626159</pre> |
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=={{header|Python}}== |
=={{header|Python}}== |
Revision as of 18:35, 22 January 2015
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
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
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
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 fraction 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
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 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 @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", .perl given beating-probability < 9d4 6d6 >; say .gist, "\t", .perl 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.
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 1 (fraction 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
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)
REXX
Algorithm
(adapted for Classic Rexx) <lang 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
Result from 100.000 tries.
<lang rexx>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=100000
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 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