Monty Hall problem

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Task
Monty Hall problem
You are encouraged to solve this task according to the task description, using any language you may know.

Run random simulations of the Monty Hall game. Show the effects of a strategy of the contestant always keeping his first guess so it can be contrasted with the strategy of the contestant always switching his guess.

Suppose you're on a game show and you're given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you "Do you want to switch to Door Number 2?" Is it to your advantage to change your choice? (Krauss and Wang 2003:10)

Note that the player may initially choose any of the three doors (not just Door 1), that the host opens a different door revealing a goat (not necessarily Door 3), and that he gives the player a second choice between the two remaining unopened doors.

Simulate at least a thousand games using three doors for each strategy and show the results in such a way as to make it easy to compare the effects of each strategy.

Reference

Contents

[edit] ActionScript

package {
import flash.display.Sprite;
 
public class MontyHall extends Sprite
{
public function MontyHall()
{
var iterations:int = 30000;
var switchWins:int = 0;
var stayWins:int = 0;
 
for (var i:int = 0; i < iterations; i++)
{
var doors:Array = [0, 0, 0];
doors[Math.floor(Math.random() * 3)] = 1;
var choice:int = Math.floor(Math.random() * 3);
var shown:int;
 
do
{
shown = Math.floor(Math.random() * 3);
} while (doors[shown] == 1 || shown == choice);
 
stayWins += doors[choice];
switchWins += doors[3 - choice - shown];
}
 
trace("Switching wins " + switchWins + " times. (" + (switchWins / iterations) * 100 + "%)");
trace("Staying wins " + stayWins + " times. (" + (stayWins / iterations) * 100 + "%)");
}
}
}

Output:

Switching wins 18788 times. (62.626666666666665%)
Staying wins 11212 times. (37.37333333333333%)

[edit] Ada

-- Monty Hall Game
 
with Ada.Text_Io; use Ada.Text_Io;
with Ada.Float_Text_Io; use Ada.Float_Text_Io;
with ada.Numerics.Discrete_Random;
 
procedure Monty_Stats is
Num_Iterations : Positive := 100000;
type Action_Type is (Stay, Switch);
type Prize_Type is (Goat, Pig, Car);
type Door_Index is range 1..3;
package Random_Prize is new Ada.Numerics.Discrete_Random(Door_Index);
use Random_Prize;
Seed : Generator;
Doors : array(Door_Index) of Prize_Type;
 
procedure Set_Prizes is
Prize_Index : Door_Index;
Booby_Prize : Prize_Type := Goat;
begin
Reset(Seed);
Prize_Index := Random(Seed);
Doors(Prize_Index) := Car;
for I in Doors'range loop
if I /= Prize_Index then
Doors(I) := Booby_Prize;
Booby_Prize := Prize_Type'Succ(Booby_Prize);
end if;
end loop;
end Set_Prizes;
 
function Play(Action : Action_Type) return Prize_Type is
Chosen : Door_Index := Random(Seed);
Monty : Door_Index;
begin
Set_Prizes;
for I in Doors'range loop
if I /= Chosen and Doors(I) /= Car then
Monty := I;
end if;
end loop;
if Action = Switch then
for I in Doors'range loop
if I /= Monty and I /= Chosen then
Chosen := I;
exit;
end if;
end loop;
end if;
return Doors(Chosen);
end Play;
Winners : Natural;
Pct : Float;
begin
Winners := 0;
for I in 1..Num_Iterations loop
if Play(Stay) = Car then
Winners := Winners + 1;
end if;
end loop;
Put("Stay : count" & Natural'Image(Winners) & " = ");
Pct := Float(Winners * 100) / Float(Num_Iterations);
Put(Item => Pct, Aft => 2, Exp => 0);
Put_Line("%");
Winners := 0;
for I in 1..Num_Iterations loop
if Play(Switch) = Car then
Winners := Winners + 1;
end if;
end loop;
Put("Switch : count" & Natural'Image(Winners) & " = ");
Pct := Float(Winners * 100) / Float(Num_Iterations);
Put(Item => Pct, Aft => 2, Exp => 0);
Put_Line("%");
 
end Monty_Stats;

Results

Stay : count 34308 = 34.31%
Switch : count 65695 = 65.69%

[edit] ALGOL 68

Translation of: C
Works with: ALGOL 68 version Standard - no extensions to language used
Works with: ALGOL 68G version Any - tested with release mk15-0.8b.fc9.i386
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8.8d.fc9.i386
INT trials=100 000;
 
PROC brand = (INT n)INT: 1 + ENTIER (n * random);
 
PROC percent = (REAL x)STRING: fixed(100.0*x/trials,0,2)+"%";
 
main:
(
INT prize, choice, show, not shown, new choice;
INT stay winning:=0, change winning:=0, random winning:=0;
INT doors = 3;
[doors-1]INT other door;
 
TO trials DO
# put the prize somewhere #
prize := brand(doors);
# let the user choose a door #
choice := brand(doors);
# let us take a list of unchoosen doors #
INT k := LWB other door;
FOR j TO doors DO
IF j/=choice THEN other door[k] := j; k+:=1 FI
OD;
# Monty opens one... #
IF choice = prize THEN
# staying the user will win... Monty opens a random port#
show := other door[ brand(doors - 1) ];
not shown := other door[ (show+1) MOD (doors - 1 ) + 1]
ELSE # no random, Monty can open just one door... #
IF other door[1] = prize THEN
show := other door[2];
not shown := other door[1]
ELSE
show := other door[1];
not shown := other door[2]
FI
FI;
 
# the user randomly choose one of the two closed doors
(one is his/her previous choice, the second is the
one not shown ) #

other door[1] := choice;
other door[2] := not shown;
new choice := other door[ brand(doors - 1) ];
# now let us count if it takes it or not #
IF choice = prize THEN stay winning+:=1 FI;
IF not shown = prize THEN change winning+:=1 FI;
IF new choice = prize THEN random winning+:=1 FI
OD;
 
print(("Staying: ", percent(stay winning), new line ));
print(("Changing: ", percent(change winning), new line ));
print(("New random choice: ", percent(random winning), new line ))
)

Sample output:

Staying: 33.62%
Changing: 66.38%
New random choice: 50.17%

[edit] APL

     ∇ Run runs;doors;i;chosen;cars;goats;swap;stay;ix;prices
[1] ⍝0: Monthy Hall problem
[2] ⍝1: http://rosettacode.org/wiki/Monty_Hall_problem
[3]
[4] (⎕IO ⎕ML)←0 1
[5] prices←0 0 1 ⍝ 0=Goat, 1=Car
[6]
[7] ix←⊃,/{3?3}¨⍳runs ⍝ random indexes of doors (placement of car)
[8] doors←(runs 3)⍴prices[ix] ⍝ matrix of doors
[9] stay←+⌿doors[;?3] ⍝ chose randomly one door - is it a car?
[10] swap←runs-stay ⍝ If not, then the other one is!
[11]
[12] ⎕←'Swap: ',(2⍕100×(swap÷runs)),'% it''s a car'
[13] ⎕←'Stay: ',(2⍕100×(stay÷runs)),'% it''s a car'
      Run 100000
Swap:  66.54% it's a car
Stay:  33.46% it's a car

[edit] AutoHotkey

#NoTrayIcon
#SingleInstance, OFF
#Persistent
SetBatchLines, -1
Iterations = 1000
Loop, %Iterations%
{
If Monty_Hall(1)
Correct_Change++
Else
Incorrect_Change++
If Monty_Hall(2)
Correct_Random++
Else
Incorrect_Random++
If Monty_Hall(3)
Correct_Stay++
Else
Incorrect_Stay++
}
Percent_Change := floor(Correct_Change / Iterations * 100)
Percent_Random := floor(Correct_Random / Iterations * 100)
Percent_Stay := floor(Correct_Stay / Iterations * 100)
MsgBox,, Monty Hall Problem, These are the results:`r`n`r`nWhen I changed my guess, I got %Correct_Change% of %Iterations% (that's %Incorrect_Change% incorrect). Thats %Percent_Change%`% correct.`r`nWhen I randomly changed my guess, I got %Correct_Random% of %Iterations% (that's %Incorrect_Random% incorrect). Thats %Percent_Random%`% correct.`r`nWhen I stayed with my first guess, I got %Correct_Stay% of %Iterations% (that's %Incorrect_Stay% incorrect). Thats %Percent_Stay%`% correct.
ExitApp
Monty_Hall(Mode) ;Mode is 1 for change, 2 for random, or 3 for stay
{
Random, prize, 1, 3
Random, guess, 1, 3
If (prize = guess && Mode != 3)
While show != 0 && show != guess
Random, show, 1, 3
Else
show := 6 - prize - guess
Random, change_guess, 0, 1
If (Mode = 1 || (change_guess && Mode = 2))
Return, (6 - show - guess) = prize
Else If (Mode = 3 || (!change_guess && Mode = 2))
Return, guess = prize
Else
Return
}

Sample output:

These are the results:

When I changed my guess, I got 762 of 1000 (that's 238 incorrect). Thats 76% correct.
When I randomly changed my guess, I got 572 of 1000 (that's 428 incorrect). Thats 57% correct.
When I stayed with my first guess, I got 329 of 1000 (that's 671 incorrect). Thats 32% correct.

[edit] AWK

#!/bin/gawk -f 
 
# Monty Hall problem
 
BEGIN {
srand()
doors = 3
iterations = 10000
# Behind a door:
EMPTY = "empty"; PRIZE = "prize"
# Algorithm used
KEEP = "keep"; SWITCH="switch"; RAND="random";
#
}
function monty_hall( choice, algorithm ) {
# Set up doors
for ( i=0; i<doors; i++ ) {
door[i] = EMPTY
}
# One door with prize
door[int(rand()*doors)] = PRIZE
 
chosen = door[choice]
del door[choice]
 
#if you didn't choose the prize first time around then
# that will be the alternative
alternative = (chosen == PRIZE) ? EMPTY : PRIZE
 
if( algorithm == KEEP) {
return chosen
}
if( algorithm == SWITCH) {
return alternative
}
return rand() <0.5 ? chosen : alternative
}
 
function simulate(algo){
prizecount = 0
for(j=0; j< iterations; j++){
if( monty_hall( int(rand()*doors), algo) == PRIZE) {
prizecount ++
}
}
printf " Algorithm %7s: prize count = %i, = %6.2f%%\n", \
algo, prizecount,prizecount*100/iterations
}
 
BEGIN {
print "\nMonty Hall problem simulation:"
print doors, "doors,", iterations, "iterations.\n"
simulate(KEEP)
simulate(SWITCH)
simulate(RAND)
 
}

Sample output:

bash$ ./monty_hall.awk
 
Monty Hall problem simulation:
3 doors, 10000 iterations.
 
Algorithm keep: prize count = 3411, = 34.11%
Algorithm switch: prize count = 6655, = 66.55%
Algorithm random: prize count = 4991, = 49.91%
bash$

[edit] BASIC

Works with: QuickBasic version 4.5
Translation of: Java
RANDOMIZE TIMER
DIM doors(3) '0 is a goat, 1 is a car
CLS
switchWins = 0
stayWins = 0
FOR plays = 0 TO 32767
winner = INT(RND * 3) + 1
doors(winner) = 1'put a winner in a random door
choice = INT(RND * 3) + 1'pick a door, any door
DO
shown = INT(RND * 3) + 1
'don't show the winner or the choice
LOOP WHILE doors(shown) = 1 OR shown = choice
stayWins = stayWins + doors(choice) 'if you won by staying, count it
switchWins = switchWins + doors(3 - choice - shown) 'could have switched to win
doors(winner) = 0 'clear the doors for the next test
NEXT plays
PRINT "Switching wins"; switchWins; "times."
PRINT "Staying wins"; stayWins; "times."

Output:

Switching wins 21805 times.
Staying wins 10963 times.

[edit] BBC BASIC

      total% = 10000
FOR trial% = 1 TO total%
prize_door% = RND(3) : REM. The prize is behind this door
guess_door% = RND(3) : REM. The contestant guesses this door
IF prize_door% = guess_door% THEN
REM. The contestant guessed right, reveal either of the others
reveal_door% = RND(2)
IF prize_door% = 1 reveal_door% += 1
IF prize_door% = 2 AND reveal_door% = 2 reveal_door% = 3
ELSE
REM. The contestant guessed wrong, so reveal the non-prize door
reveal_door% = prize_door% EOR guess_door%
ENDIF
stick_door% = guess_door% : REM. The sticker doesn't change his mind
swap_door% = guess_door% EOR reveal_door% : REM. but the swapper does
IF stick_door% = prize_door% sticker% += 1
IF swap_door% = prize_door% swapper% += 1
NEXT trial%
PRINT "After a total of ";total%;" trials,"
PRINT "The 'sticker' won ";sticker%;" times (";INT(sticker%/total%*100);"%)"
PRINT "The 'swapper' won ";swapper%;" times (";INT(swapper%/total%*100);"%)"

Output:

After a total of 10000 trials,
The 'sticker' won 3379 times (33%)
The 'swapper' won 6621 times (66%)

[edit] C

//Evidence of the Monty Hall solution.
 
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
 
#define GAMES 3000000
 
int main(void){
unsigned i, j, k, choice, winsbyswitch=0, door[3];
 
srand(time(NULL)); //initialize random seed.
for(i=0; i<GAMES; i++){
door[0] = (!(rand()%2)) ? 1: 0; //give door 1 either a car or a goat randomly.
if(door[0]) door[1]=door[2]=0; //if 1st door has car, give other doors goats.
else{ door[1] = (!(rand()%2)) ? 1: 0; door[2] = (!door[1]) ? 1: 0; } //else, give 2nd door car or goat, give 3rd door what's left.
choice = rand()%3; //choose a random door.
 
//if the next door has a goat, and the following door has a car, or vice versa, you'd win if you switch.
if(((!(door[((choice+1)%3)])) && (door[((choice+2)%3)])) || (!(door[((choice+2)%3)]) && (door[((choice+1)%3)]))) winsbyswitch++;
}
printf("\nAfter %u games, I won %u by switching. That is %f%%. ", GAMES, winsbyswitch, (float)winsbyswitch*100.0/(float)i);
}
 

Output of one run:

After 3000000 games, I won 1999747 by switching.  That is 66.658233%. 

[edit] C#

Translation of: Java
using System;
 
class Program
{
static void Main(string[] args)
{
int switchWins = 0;
int stayWins = 0;
 
Random gen = new Random();
 
for(int plays = 0; plays < 1000000; plays++ )
{
int[] doors = {0,0,0};//0 is a goat, 1 is a car
 
var winner = gen.Next(3);
doors[winner] = 1; //put a winner in a random door
 
int choice = gen.Next(3); //pick a door, any door
int shown; //the shown door
do
{
shown = gen.Next(3);
}
while (doors[shown] == 1 || shown == choice); //don't show the winner or the choice
 
stayWins += doors[choice]; //if you won by staying, count it
 
//the switched (last remaining) door is (3 - choice - shown), because 0+1+2=3
switchWins += doors[3 - choice - shown];
}
 
Console.Out.WriteLine("Staying wins " + stayWins + " times.");
Console.Out.WriteLine("Switching wins " + switchWins + " times.");
}
}

Sample output:

Staying wins:    333830
Switching wins:  666170

[edit] C++

#include <iostream>
#include <cstdlib>
#include <ctime>
 
int randint(int n)
{
return (1.0*n*std::rand())/(1.0+RAND_MAX);
}
 
int other(int doorA, int doorB)
{
int doorC;
if (doorA == doorB)
{
doorC = randint(2);
if (doorC >= doorA)
++doorC;
}
else
{
for (doorC = 0; doorC == doorA || doorC == doorB; ++doorC)
{
// empty
}
}
return doorC;
}
 
int check(int games, bool change)
{
int win_count = 0;
for (int game = 0; game < games; ++game)
{
int const winning_door = randint(3);
int const original_choice = randint(3);
int open_door = other(original_choice, winning_door);
 
int const selected_door = change?
other(open_door, original_choice)
: original_choice;
 
if (selected_door == winning_door)
++win_count;
}
 
return win_count;
}
 
int main()
{
std::srand(std::time(0));
 
int games = 10000;
int wins_stay = check(games, false);
int wins_change = check(games, true);
std::cout << "staying: " << 100.0*wins_stay/games << "%, changing: " << 100.0*wins_change/games << "%\n";
}

Sample output:

staying: 33.73%, changing: 66.9%

[edit] Clojure

(ns monty-hall-problem
(:use [clojure.contrib.seq :only (shuffle)]))
 
(defn play-game [staying]
(let [doors (shuffle [:goat :goat :car])
choice (rand-int 3)
[a b] (filter #(not= choice %) (range 3))
alternative (if (= :goat (nth doors a)) b a)]
(= :car (nth doors (if staying choice alternative)))))
 
(defn simulate [staying times]
(let [wins (reduce (fn [counter _] (if (play-game staying) (inc counter) counter))
0
(range times))]
(str "wins " wins " times out of " times)))
 
monty-hall-problem> (println "staying:" (simulate true 1000))
staying: wins 337 times out of 1000
nil
monty-hall-problem> (println "switching:" (simulate false 1000))
switching: wins 638 times out of 1000
nil
 

[edit] COBOL

Works with: OpenCOBOL
       IDENTIFICATION DIVISION.
PROGRAM-ID. monty-hall.
 
DATA DIVISION.
WORKING-STORAGE SECTION.
78 Num-Games VALUE 1000000.
 
*> These are needed so the values are passed to
*> get-rand-int correctly.
01 One PIC 9 VALUE 1.
01 Three PIC 9 VALUE 3.
 
01 doors-area.
03 doors PIC 9 OCCURS 3 TIMES.
 
01 choice PIC 9.
01 shown PIC 9.
01 winner PIC 9.
 
01 switch-wins PIC 9(7).
01 stay-wins PIC 9(7).
 
01 stay-wins-percent PIC Z9.99.
01 switch-wins-percent PIC Z9.99.
 
PROCEDURE DIVISION.
PERFORM Num-Games TIMES
MOVE 0 TO doors (winner)
 
CALL "get-rand-int" USING CONTENT One, Three,
REFERENCE winner
MOVE 1 TO doors (winner)
 
CALL "get-rand-int" USING CONTENT One, Three,
REFERENCE choice
 
PERFORM WITH TEST AFTER
UNTIL NOT(shown = winner OR choice)
CALL "get-rand-int" USING CONTENT One, Three,
REFERENCE shown
END-PERFORM
 
ADD doors (choice) TO stay-wins
ADD doors (6 - choice - shown) TO switch-wins
END-PERFORM
 
COMPUTE stay-wins-percent ROUNDED =
stay-wins / Num-Games * 100
COMPUTE switch-wins-percent ROUNDED =
switch-wins / Num-Games * 100
 
DISPLAY "Staying wins " stay-wins " times ("
stay-wins-percent "%)."
DISPLAY "Switching wins " switch-wins " times ("
switch-wins-percent "%)."
.
 
IDENTIFICATION DIVISION.
PROGRAM-ID. get-rand-int.
 
DATA DIVISION.
WORKING-STORAGE SECTION.
01 call-flag PIC X VALUE "Y".
88 first-call VALUE "Y", FALSE "N".
 
01 num-range PIC 9.
 
LINKAGE SECTION.
01 min-num PIC 9.
01 max-num PIC 9.
 
01 ret PIC 9.
 
PROCEDURE DIVISION USING min-num, max-num, ret.
*> Seed RANDOM once.
IF first-call
MOVE FUNCTION RANDOM(FUNCTION CURRENT-DATE (9:8))
TO num-range
SET first-call TO FALSE
END-IF
 
COMPUTE num-range = max-num - min-num + 1
COMPUTE ret =
FUNCTION MOD(FUNCTION RANDOM * 100000, num-range)
+ min-num
.
END PROGRAM get-rand-int.
 
END PROGRAM monty-hall.
Output:
Staying wins   0333396 times (33.34%).
Switching wins 0666604 times (66.66%).

[edit] ColdFusion

<cfscript>
function runmontyhall(num_tests) {
// number of wins when player switches after original selection
switch_wins = 0;
// number of wins when players "sticks" with original selection
stick_wins = 0;
// run all the tests
for(i=1;i<=num_tests;i++) {
// unconditioned potential for selection of each door
doors = [0,0,0];
// winning door is randomly assigned...
winner = randrange(1,3);
// ...and actualized in the array of real doors
doors[winner] = 1;
// player chooses one of three doors
choice = randrange(1,3);
do {
// monty randomly reveals a door...
shown = randrange(1,3);
}
// ...but monty only reveals empty doors;
// he will not reveal the door that the player has choosen
// nor will he reveal the winning door
while(shown==choice || doors[shown]==1);
// when the door the player originally selected is the winner, the "stick" option gains a point
stick_wins += doors[choice];
// to calculate the number of times the player would have won with a "switch", subtract the
// "value" of the chosen, "stuck-to" door from 1, the possible number of wins if the player
// chose and stuck with the winning door (1), the player would not have won by switching, so
// the value is 1-1=0 if the player chose and stuck with a losing door (0), the player would
// have won by switching, so the value is 1-0=1
switch_wins += 1-doors[choice];
}
// finally, simply run the percentages for each outcome
stick_percentage = (stick_wins/num_tests)*100;
switch_percentage = (switch_wins/num_tests)*100;
writeoutput('Number of Tests: ' & num_tests);
writeoutput('<br />Stick Wins: ' & stick_wins & ' ['& stick_percentage &'%]');
writeoutput('<br />Switch Wins: ' & switch_wins & ' ['& switch_percentage &'%]');
}
runmontyhall(10000);
</cfscript>

Output:

Tests: 10,000  |  Switching wins:  6655 [66.55%]  |  Sticking wins:  3345 [33.45%]

[edit] Common Lisp

(defun make-round ()
(let ((array (make-array 3
:element-type 'bit
:initial-element 0)))
(setf (bit array (random 3)) 1)
array))
 
(defun show-goat (initial-choice array)
(loop for i = (random 3)
when (and (/= initial-choice i)
(zerop (bit array i)))
return i))
 
(defun won? (array i)
(= 1 (bit array i)))
CL-USER> (progn (loop repeat #1=(expt 10 6)
for round = (make-round)
for initial = (random 3)
for goat = (show-goat initial round)
for choice = (loop for i = (random 3)
when (and (/= i initial)
(/= i goat))
return i)
when (won? round (random 3))
sum 1 into result-stay
when (won? round choice)
sum 1 into result-switch
finally (progn (format t "Stay: ~S%~%" (float (/ result-stay
#1# 1/100)))
(format t "Switch: ~S%~%" (float (/ result-switch
#1# 1/100))))))
Stay: 33.2716%
Switch: 66.6593%
 
;Find out how often we win if we always switch
(defun rand-elt (s)
(elt s (random (length s))))
 
(defun monty ()
(let* ((doors '(0 1 2))
(prize (random 3));possible values: 0, 1, 2
(pick (random 3))
(opened (rand-elt (remove pick (remove prize doors))));monty opens a door which is not your pick and not the prize
(other (car (remove pick (remove opened doors))))) ;you decide to switch to the one other door that is not your pick and not opened
(= prize other))) ; did you switch to the prize?
 
(defun monty-trials (n)
(count t (loop for x from 1 to n collect (monty))))
 

[edit] D

import std.stdio, std.random;
 
void main() {
int switchWins, stayWins;
 
while (switchWins + stayWins < 100_000) {
immutable carPos = uniform(0, 3); // Which door is car behind?
immutable pickPos = uniform(0, 3); // Contestant's initial pick.
int openPos; // Which door is opened by Monty Hall?
 
// Monty can't open the door you picked or the one with the car
// behind it.
do {
openPos = uniform(0, 3);
} while(openPos == pickPos || openPos == carPos);
 
int switchPos;
// Find position that's not currently picked by contestant and
// was not opened by Monty already.
for (; pickPos==switchPos || openPos==switchPos; switchPos++) {}
 
if (pickPos == carPos)
stayWins++;
else if (switchPos == carPos)
switchWins++;
else
assert(0); // Can't happen.
}
 
writefln("Switching/Staying wins: %d %d", switchWins, stayWins);
}
Output:
Switching/Staying wins: 66609 33391

[edit] Dart

The class Game attempts to hide the implementation as much as possible, the play() function does not use any specifics of the implementation.

int rand(int max) => (Math.random()*max).toInt();
 
class Game {
int _prize;
int _open;
int _chosen;
 
Game() {
_prize=rand(3);
_open=null;
_chosen=null;
}
 
void choose(int door) {
_chosen=door;
}
 
void reveal() {
if(_prize==_chosen) {
int toopen=rand(2);
if (toopen>=_prize)
toopen++;
_open=toopen;
} else {
for(int i=0;i<3;i++)
if(_prize!=i && _chosen!=i) {
_open=i;
break;
}
}
}
 
void change() {
for(int i=0;i<3;i++)
if(_chosen!=i && _open!=i) {
_chosen=i;
break;
}
}
 
bool hasWon() => _prize==_chosen;
 
String toString() {
String res="Prize is behind door $_prize";
if(_chosen!=null) res+=", player has chosen door $_chosen";
if(_open!=null) res+=", door $_open is open";
return res;
}
}
 
void play(int count, bool swap) {
int wins=0;
 
for(int i=0;i<count;i++) {
Game game=new Game();
game.choose(rand(3));
game.reveal();
if(swap)
game.change();
if(game.hasWon())
wins++;
}
String withWithout=swap?"with":"without";
double percent=(wins*100.0)/count;
print("playing $withWithout switching won $percent%");
}
 
test() {
for(int i=0;i<5;i++) {
Game g=new Game();
g.choose(i%3);
g.reveal();
print(g);
g.change();
print(g);
print("win==${g.hasWon()}");
}
}
 
main() {
play(10000,false);
play(10000,true);
}
playing without switching won 33.32%
playing with switching won 67.63%

[edit] Eiffel

 
note
description: "[
Monty Hall Problem as an Eiffel Solution
 
1. Set the stage: Randomly place car and two goats behind doors 1, 2 and 3.
2. Monty offers choice of doors --> Contestant will choose a random door or always one door.
2a. Door has Goat - door remains closed
2b. Door has Car - door remains closed
3. Monty offers cash --> Contestant takes or refuses cash.
3a. Takes cash: Contestant is Cash winner and door is revealed. Car Loser if car door revealed.
3b. Refuses cash: Leads to offer to switch doors.
4. Monty offers door switch --> Contestant chooses to stay or change.
5. Door reveal: Contestant refused cash and did or did not door switch. Either way: Reveal!
6. Winner and Loser based on door reveal of prize.
 
Car Winner: Chooses car door
Cash Winner: Chooses cash over any door
Goat Loser: Chooses goat door
Car Loser: Chooses cash over car door or switches from car door to goat door
]"

date: "$Date$"
revision: "$Revision$"
 
class
MH_APPLICATION
 
create
make
 
feature {NONE} -- Initialization
 
make
-- Initialize Current.
do
play_lets_make_a_deal
ensure
played_1000_games: game_count = times_to_play
end
 
feature {NONE} -- Implementation: Access
 
live_contestant: attached like contestant
-- Attached version of `contestant'
do
if attached contestant as al_contestant then
Result := al_contestant
else
create Result
check not_attached_contestant: False end
end
end
 
contestant: detachable TUPLE [first_door_choice, second_door_choice: like door_number_anchor; takes_cash, switches_door: BOOLEAN]
-- Contestant for Current.
 
active_stage_door (a_door: like door_anchor): attached like door_anchor
-- Attached version of `a_door'.
do
if attached a_door as al_door then
Result := al_door
else
create Result
check not_attached_door: False end
end
end
 
door_1, door_2, door_3: like door_anchor
-- Doors with prize names and flags for goat and open (revealed).
 
feature {NONE} -- Implementation: Status
 
game_count, car_win_count, cash_win_count, car_loss_count, goat_loss_count, goat_avoidance_count: like counter_anchor
switch_count, switch_win_count: like counter_anchor
no_switch_count, no_switch_win_count: like counter_anchor
-- Counts of games played, wins and losses based on car, cash or goat.
 
feature {NONE} -- Implementation: Basic Operations
 
prepare_stage
-- Prepare the stage in terms of what doors have what prizes.
do
inspect new_random_of (3)
when 1 then
door_1 := door_with_car
door_2 := door_with_goat
door_3 := door_with_goat
when 2 then
door_1 := door_with_goat
door_2 := door_with_car
door_3 := door_with_goat
when 3 then
door_1 := door_with_goat
door_2 := door_with_goat
door_3 := door_with_car
end
active_stage_door (door_1).number := 1
active_stage_door (door_2).number := 2
active_stage_door (door_3).number := 3
ensure
door_has_prize: not active_stage_door (door_1).is_goat or
not active_stage_door (door_2).is_goat or
not active_stage_door (door_3).is_goat
consistent_door_numbers: active_stage_door (door_1).number = 1 and
active_stage_door (door_2).number = 2 and
active_stage_door (door_3).number = 3
end
 
door_number_having_prize: like door_number_anchor
-- What door number has the car?
do
if not active_stage_door (door_1).is_goat then
Result := 1
elseif not active_stage_door (door_2).is_goat then
Result := 2
elseif not active_stage_door (door_3).is_goat then
Result := 3
else
check prize_not_set: False end
end
ensure
one_to_three: between_1_and_x_inclusive (3, Result)
end
 
door_with_car: attached like door_anchor
-- Create a door with a car.
do
create Result
Result.name := prize
ensure
not_empty: not Result.name.is_empty
name_is_prize: Result.name.same_string (prize)
end
 
door_with_goat: attached like door_anchor
-- Create a door with a goat
do
create Result
Result.name := gag_gift
Result.is_goat := True
ensure
not_empty: not Result.name.is_empty
name_is_prize: Result.name.same_string (gag_gift)
is_gag_gift: Result.is_goat
end
 
next_contestant: attached like live_contestant
-- The next contestant on Let's Make a Deal!
do
create Result
Result.first_door_choice := new_random_of (3)
Result.second_door_choice := choose_another_door (Result.first_door_choice)
Result.takes_cash := random_true_or_false
if not Result.takes_cash then
Result.switches_door := random_true_or_false
end
ensure
choices_one_to_three: Result.first_door_choice <= 3 and Result.second_door_choice <= 3
switch_door_implies_no_cash_taken: Result.switches_door implies not Result.takes_cash
end
 
choose_another_door (a_first_choice: like door_number_anchor): like door_number_anchor
-- Make a choice from the remaining doors
require
one_to_three: between_1_and_x_inclusive (3, a_first_choice)
do
Result := new_random_of (3)
from until Result /= a_first_choice
loop
Result := new_random_of (3)
end
ensure
first_choice_not_second: a_first_choice /= Result
result_one_to_three: between_1_and_x_inclusive (3, Result)
end
 
play_lets_make_a_deal
-- Play the game 1000 times
local
l_car_win, l_car_loss, l_cash_win, l_goat_loss, l_goat_avoided: BOOLEAN
do
from
game_count := 0
invariant
consistent_win_loss_counts: (game_count = (car_win_count + cash_win_count + goat_loss_count))
consistent_loss_avoidance_counts: (game_count = (car_loss_count + goat_avoidance_count))
until
game_count >= times_to_play
loop
prepare_stage
contestant := next_contestant
l_cash_win := (live_contestant.takes_cash)
 
l_car_win := (not l_cash_win and
(not live_contestant.switches_door and live_contestant.first_door_choice = door_number_having_prize) or
(live_contestant.switches_door and live_contestant.second_door_choice = door_number_having_prize))
 
l_car_loss := (not live_contestant.switches_door and live_contestant.first_door_choice /= door_number_having_prize) or
(live_contestant.switches_door and live_contestant.second_door_choice /= door_number_having_prize)
 
l_goat_loss := (not l_car_win and not l_cash_win)
 
l_goat_avoided := (not live_contestant.switches_door and live_contestant.first_door_choice = door_number_having_prize) or
(live_contestant.switches_door and live_contestant.second_door_choice = door_number_having_prize)
 
check consistent_goats: l_goat_loss implies not l_goat_avoided end
check consistent_car_win: l_car_win implies not l_car_loss and not l_cash_win and not l_goat_loss end
check consistent_cash_win: l_cash_win implies not l_car_win and not l_goat_loss end
check consistent_goat_avoidance: l_goat_avoided implies (l_car_win or l_cash_win) and not l_goat_loss end
check consistent_car_loss: l_car_loss implies l_cash_win or l_goat_loss end
 
if l_car_win then car_win_count := car_win_count + 1 end
if l_cash_win then cash_win_count := cash_win_count + 1 end
if l_goat_loss then goat_loss_count := goat_loss_count + 1 end
if l_car_loss then car_loss_count := car_loss_count + 1 end
if l_goat_avoided then goat_avoidance_count := goat_avoidance_count + 1 end
 
if live_contestant.switches_door then
switch_count := switch_count + 1
if l_car_win then
switch_win_count := switch_win_count + 1
end
else -- if not live_contestant.takes_cash and not live_contestant.switches_door then
no_switch_count := no_switch_count + 1
if l_car_win or l_cash_win then
no_switch_win_count := no_switch_win_count + 1
end
end
 
 
game_count := game_count + 1
end
print ("%NCar Wins:%T%T " + car_win_count.out +
"%NCash Wins:%T%T " + cash_win_count.out +
"%NGoat Losses:%T%T " + goat_loss_count.out +
"%N-----------------------------" +
"%NTotal Win/Loss:%T%T" + (car_win_count + cash_win_count + goat_loss_count).out +
"%N%N" +
"%NCar Losses:%T%T " + car_loss_count.out +
"%NGoats Avoided:%T%T " + goat_avoidance_count.out +
"%N-----------------------------" +
"%NTotal Loss/Avoid:%T" + (car_loss_count + goat_avoidance_count).out +
"%N-----------------------------" +
"%NStaying Count/Win:%T" + no_switch_count.out + "/" + no_switch_win_count.out + " = " + (no_switch_win_count / no_switch_count * 100).out + " %%" +
"%NSwitch Count/Win:%T" + switch_count.out + "/" + switch_win_count.out + " = " + (switch_win_count / switch_count * 100).out + " %%"
)
end
 
feature {NONE} -- Implementation: Random Numbers
 
last_random: like random_number_anchor
-- The last random number chosen.
 
random_true_or_false: BOOLEAN
-- A randome True or False
do
Result := new_random_of (2) = 2
end
 
new_random_of (a_number: like random_number_anchor): like door_number_anchor
-- A random number from 1 to `a_number'.
do
Result := (new_random \\ a_number + 1).as_natural_8
end
 
new_random: like random_number_anchor
-- Random integer
-- Each call returns another random number.
do
random_sequence.forth
Result := random_sequence.item
last_random := Result
ensure
old_random_not_new: old last_random /= last_random
end
 
random_sequence: RANDOM
-- Random sequence seeded from clock when called.
attribute
create Result.set_seed ((create {TIME}.make_now).milli_second)
end
 
feature {NONE} -- Implementation: Constants
 
times_to_play: NATURAL_16 = 1000
-- Times to play the game.
 
prize: STRING = "Car"
-- Name of the prize
 
gag_gift: STRING = "Goat"
-- Name of the gag gift
 
door_anchor: detachable TUPLE [number: like door_number_anchor; name: STRING; is_goat, is_open: BOOLEAN]
-- Type anchor for door tuples.
 
door_number_anchor: NATURAL_8
-- Type anchor for door numbers.
 
random_number_anchor: INTEGER
-- Type anchor for random numbers.
 
counter_anchor: NATURAL_16
-- Type anchor for counters.
 
feature {NONE} -- Implementation: Contract Support
 
between_1_and_x_inclusive (a_number, a_value: like door_number_anchor): BOOLEAN
-- Is `a_value' between 1 and `a_number'?
do
Result := (a_value > 0) and (a_value <= a_number)
end
 
end
 
Output:
Car Wins:                177
Cash Wins:               486
Goat Losses:             337
-----------------------------
Total Win/Loss:         1000


Car Losses:              657
Goats Avoided:           343
-----------------------------
Total Loss/Avoid:       1000
-----------------------------
Staying Count/Win:      742/573 = 77.223719676549862 %
Switch  Count/Win:      258/90 = 34.883720930232556 %


[edit] Emacs Lisp

Translation of: Picolisp
 
(defun montyhall (keep)
(let
((prize (random 3))
(choice (random 3)))
(if keep (= prize choice)
(/= prize choice))))
 
 
(let ((cnt 0))
(dotimes (i 10000)
(and (montyhall t) (setq cnt (1+ cnt))))
(princ (format "Strategy keep: %.3f %%" (/ cnt 100.0))))
 
(let ((cnt 0))
(dotimes (i 10000)
(and (montyhall nil) (setq cnt (1+ cnt))))
(princ (format "Strategy switch: %.3f %%" (/ cnt 100.0))))
 
Output:
Strategy keep: 34.410 %
Strategy switch: 66.430 %


[edit] Erlang

-module(monty_hall).
 
-export([main/0]).
 
main() ->
random:seed(now()),
{WinStay, WinSwitch} = experiment(100000, 0, 0),
io:format("Switching wins ~p times.\n", [WinSwitch]),
io:format("Staying wins ~p times.\n", [WinStay]).
 
experiment(0, WinStay, WinSwitch) ->
{WinStay, WinSwitch};
experiment(N, WinStay, WinSwitch) ->
Doors = setelement(random:uniform(3), {0,0,0}, 1),
SelectedDoor = random:uniform(3),
OpenDoor = open_door(Doors, SelectedDoor),
experiment(
N - 1,
WinStay + element(SelectedDoor, Doors),
WinSwitch + element(6 - (SelectedDoor + OpenDoor), Doors) ).
 
open_door(Doors,SelectedDoor) ->
OpenDoor = random:uniform(3),
case (element(OpenDoor, Doors) =:= 1) or (OpenDoor =:= SelectedDoor) of
true -> open_door(Doors, SelectedDoor);
false -> OpenDoor
end.
 

Sample Output:

Switching wins 66595 times.
Staying wins 33405 times.

[edit] Euphoria

integer switchWins, stayWins
switchWins = 0
stayWins = 0
 
integer winner, choice, shown
 
for plays = 1 to 10000 do
winner = rand(3)
choice = rand(3)
while 1 do
shown = rand(3)
if shown != winner and shown != choice then
exit
end if
end while
stayWins += choice = winner
switchWins += 6-choice-shown = winner
end for
printf(1, "Switching wins %d times\n", switchWins)
printf(1, "Staying wins %d times\n", stayWins)
 

Sample Output:

Switching wins 6697 times
Staying wins 3303 times

[edit] F#

I don't bother with having Monty "pick" a door, since you only win if you initially pick a loser in the switch strategy and you only win if you initially pick a winner in the stay strategy so there doesn't seem to be much sense in playing around the background having Monty "pick" doors. Makes it pretty simple to see why it's always good to switch.

open System
let monty nSims =
let rnd = new Random()
let SwitchGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
if winner <> pick then 1 else 0
 
let StayGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
if winner = pick then 1 else 0
 
let Wins (f:unit -> int) = seq {for i in [1..nSims] -> f()} |> Seq.sum
printfn "Stay: %d wins out of %d - Switch: %d wins out of %d" (Wins StayGame) nSims (Wins SwitchGame) nSims

Sample Output:

Stay: 332874 wins out of 1000000 - Switch: 667369 wins out of 1000000

I had a very polite suggestion that I simulate Monty's "pick" so I'm putting in a version that does that. I compare the outcome with my original outcome and, unsurprisingly, show that this is essentially a noop that has no bearing on the output, but I (kind of) get where the request is coming from so here's that version...

let montySlower nSims =
let rnd = new Random()
let MontyPick winner pick =
if pick = winner then
[0..2] |> Seq.filter (fun i -> i <> pick) |> Seq.nth (rnd.Next(0,2))
else
3 - pick - winner
let SwitchGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
let monty = MontyPick winner pick
let pickFinal = 3 - monty - pick
 
// Show that Monty's pick has no effect...
 
if (winner <> pick) <> (pickFinal = winner) then
printfn "Monty's selection actually had an effect!"
if pickFinal = winner then 1 else 0
 
let StayGame() =
let winner, pick = rnd.Next(0,3), rnd.Next(0,3)
let monty = MontyPick winner pick
 
// This one's even more obvious than the above since pickFinal
// is precisely the same as pick
 
let pickFinal = pick
if (winner = pick) <> (winner = pickFinal) then
printfn "Monty's selection actually had an effect!"
if winner = pickFinal then 1 else 0
 
let Wins (f:unit -> int) = seq {for i in [1..nSims] -> f()} |> Seq.sum
printfn "Stay: %d wins out of %d - Switch: %d wins out of %d" (Wins StayGame) nSims (Wins SwitchGame) nSims

[edit] Forth

include random.fs
 
variable stay-wins
variable switch-wins
 
: trial ( -- )
3 random 3 random ( prize choice )
= if 1 stay-wins +!
else 1 switch-wins +!
then ;
: trials ( n -- )
0 stay-wins ! 0 switch-wins !
dup 0 do trial loop
cr stay-wins @ . [char] / emit dup . ." staying wins"
cr switch-wins @ . [char] / emit . ." switching wins" ;
 
1000 trials

or in iForth:

0 value stay-wins
0 value switch-wins
 
: trial ( -- )
3 choose 3 choose ( -- prize choice )
= IF 1 +TO stay-wins exit ENDIF
1 +TO switch-wins ;
 
: trials ( n -- )
CLEAR stay-wins
CLEAR switch-wins
dup 0 ?DO trial LOOP
CR stay-wins DEC. ." / " dup DEC. ." staying wins,"
CR switch-wins DEC. ." / " DEC. ." switching wins." ;

With output:

 FORTH> 100000000 trials
 33336877 / 100000000 staying wins,
 66663123 / 100000000 switching wins. ok

[edit] Fortran

Works with: Fortran version 90 and later
PROGRAM MONTYHALL
 
IMPLICIT NONE
 
INTEGER, PARAMETER :: trials = 10000
INTEGER :: i, choice, prize, remaining, show, staycount = 0, switchcount = 0
LOGICAL :: door(3)
REAL :: rnum
 
CALL RANDOM_SEED
DO i = 1, trials
door = .FALSE.
CALL RANDOM_NUMBER(rnum)
prize = INT(3*rnum) + 1
door(prize) = .TRUE. ! place car behind random door
 
CALL RANDOM_NUMBER(rnum)
choice = INT(3*rnum) + 1 ! choose a door
 
DO
CALL RANDOM_NUMBER(rnum)
show = INT(3*rnum) + 1
IF (show /= choice .AND. show /= prize) EXIT ! Reveal a goat
END DO
 
SELECT CASE(choice+show) ! Calculate remaining door index
CASE(3)
remaining = 3
CASE(4)
remaining = 2
CASE(5)
remaining = 1
END SELECT
 
IF (door(choice)) THEN ! You win by staying with your original choice
staycount = staycount + 1
ELSE IF (door(remaining)) THEN ! You win by switching to other door
switchcount = switchcount + 1
END IF
 
END DO
 
WRITE(*, "(A,F6.2,A)") "Chance of winning by not switching is", real(staycount)/trials*100, "%"
WRITE(*, "(A,F6.2,A)") "Chance of winning by switching is", real(switchcount)/trials*100, "%"
 
END PROGRAM MONTYHALL

Sample Output

Chance of winning by not switching is 32.82%
Chance of winning by switching is 67.18%

[edit] Go

package main
 
import (
"fmt"
"math/rand"
"time"
)
 
func main() {
games := 100000
r := rand.New(rand.NewSource(time.Now().UnixNano()))
 
var switcherWins, keeperWins, shown int
for i := 0; i < games; i++ {
doors := []int{0, 0, 0}
doors[r.Intn(3)] = 1 // Set which one has the car
choice := r.Intn(3) // Choose a door
for shown = r.Intn(3); shown == choice || doors[shown] == 1; shown = r.Intn(3) {}
switcherWins += doors[3 - choice - shown]
keeperWins += doors[choice]
}
floatGames := float32(games)
fmt.Printf("Switcher Wins: %d (%3.2f%%)\n",
switcherWins, (float32(switcherWins) / floatGames * 100))
fmt.Printf("Keeper Wins: %d (%3.2f%%)",
keeperWins, (float32(keeperWins) / floatGames * 100))
}

Output:

Switcher Wins: 66542 (66.54%)
Keeper Wins: 33458 (33.46%)

[edit] Haskell

import System.Random (StdGen, getStdGen, randomR)
 
trials :: Int
trials = 10000
 
data Door = Car | Goat deriving Eq
 
play :: Bool -> StdGen -> (Door, StdGen)
play switch g = (prize, new_g)
where (n, new_g) = randomR (0, 2) g
d1 = [Car, Goat, Goat] !! n
prize = case switch of
False -> d1
True -> case d1 of
Car -> Goat
Goat -> Car
 
cars :: Int -> Bool -> StdGen -> (Int, StdGen)
cars n switch g = f n (0, g)
where f 0 (cs, g) = (cs, g)
f n (cs, g) = f (n - 1) (cs + result, new_g)
where result = case prize of Car -> 1; Goat -> 0
(prize, new_g) = play switch g
 
main = do
g <- getStdGen
let (switch, g2) = cars trials True g
(stay, _) = cars trials False g2
putStrLn $ msg "switch" switch
putStrLn $ msg "stay" stay
where msg strat n = "The " ++ strat ++ " strategy succeeds " ++
percent n ++ "% of the time."
percent n = show $ round $
100 * (fromIntegral n) / (fromIntegral trials)
Library: mtl

With a State monad, we can avoid having to explicitly pass around the StdGen so often. play and cars can be rewritten as follows:

import Control.Monad.State
 
play :: Bool -> State StdGen Door
play switch = do
i <- rand
let d1 = [Car, Goat, Goat] !! i
return $ case switch of
False -> d1
True -> case d1 of
Car -> Goat
Goat -> Car
where rand = do
g <- get
let (v, new_g) = randomR (0, 2) g
put new_g
return v
 
cars :: Int -> Bool -> StdGen -> (Int, StdGen)
cars n switch g = (numcars, new_g)
where numcars = length $ filter (== Car) prize_list
(prize_list, new_g) = runState (replicateM n (play switch)) g

Sample output (for either implementation):

The switch strategy succeeds 67% of the time.
The stay strategy succeeds 34% of the time.

[edit] HicEst

REAL :: ndoors=3, doors(ndoors), plays=1E4
 
DLG(NameEdit = plays, DNum=1, Button='Go')
 
switchWins = 0
stayWins = 0
 
DO play = 1, plays
doors = 0 ! clear the doors
winner = 1 + INT(RAN(ndoors)) ! door that has the prize
doors(winner) = 1
guess = 1 + INT(RAN(doors)) ! player chooses his door
 
IF( guess == winner ) THEN ! Monty decides which door to open:
show = 1 + INT(RAN(2)) ! select 1st or 2nd goat-door
checked = 0
DO check = 1, ndoors
checked = checked + (doors(check) == 0)
IF(checked == show) open = check
ENDDO
ELSE
open = (1+2+3) - winner - guess
ENDIF
new_guess_if_switch = (1+2+3) - guess - open
 
stayWins = stayWins + doors(guess) ! count if guess was correct
switchWins = switchWins + doors(new_guess_if_switch)
ENDDO
 
WRITE(ClipBoard, Name) plays, switchWins, stayWins
 
END
! plays=1E3; switchWins=695; stayWins=305;
! plays=1E4; switchWins=6673; stayWins=3327;
! plays=1E5; switchWins=66811; stayWins=33189;
! plays=1E6; switchWins=667167; stayWins=332833;

[edit] Icon and Unicon

procedure main(arglist)
 
rounds := integer(arglist[1]) | 10000
doors := '123'
strategy1 := strategy2 := 0
 
every 1 to rounds do {
goats := doors -- ( car := ?doors )
guess1 := ?doors
show := goats -- guess1
if guess1 == car then strategy1 +:= 1
else strategy2 +:= 1
}
 
write("Monty Hall simulation for ", rounds, " rounds.")
write("Strategy 1 'Staying' won ", real(strategy1) / rounds )
write("Strategy 2 'Switching' won ", real(strategy2) / rounds )
 
end
Sample Output:
Monty Hall simulation for 10000 rounds.
Strategy 1 'Staying' won 0.3266
Strategy 2 'Switching' won 0.6734

[edit] Io

keepWins := 0
switchWins := 0
doors := 3
times := 100000
pickDoor := method(excludeA, excludeB,
door := excludeA
while(door == excludeA or door == excludeB,
door = (Random value() * doors) floor
)
door
)
times repeat(
playerChoice := pickDoor()
carDoor := pickDoor()
shownDoor := pickDoor(carDoor, playerChoice)
switchDoor := pickDoor(playerChoice, shownDoor)
(playerChoice == carDoor) ifTrue(keepWins = keepWins + 1)
(switchDoor == carDoor) ifTrue(switchWins = switchWins + 1)
)
("Switching to the other door won #{switchWins} times.\n"\
.. "Keeping the same door won #{keepWins} times.\n"\
.. "Game played #{times} times with #{doors} doors.") interpolate println
 
Sample output:
Switching to the other door won 66935 times.
Keeping the same door won 33065 times.
Game played 100000 times with 3 doors.

[edit] J

The core of this simulation is picking a random item from a set

pick=: {~ ?@#

And, of course, we will be picking one door from three doors

DOORS=:1 2 3

But note that the simulation code should work just as well with more doors.

Anyways the scenario where the contestant's switch or stay strategy makes a difference is where Monty has picked from the doors which are neither the user's door nor the car's door.

scenario=: ((pick@-.,])pick,pick) bind DOORS

(Here, I have decided that the result will be a list of three door numbers. The first number in that list is the number Monty picks, the second number represents the door the user picked, and the third number represents the door where the car is hidden.)

Once we have our simulation test results for the scenario, we need to test if staying would win. In other words we need to test if the user's first choice matches where the car was hidden:

stayWin=: =/@}.

In other words: drop the first element from the list representing our test results -- this leaves us with the user's choice and the door where the car was hidden -- and then insert the verb = between those two values.

We also need to test if switching would win. In other words, we need to test if the user would pick the car from the doors other than the one Monty picked and the one the user originally picked:

switchWin=: pick@(DOORS -. }:) = {:

In other words, start with our list of all doors and then remove the door the monty picked and the door the user picked, and then pick one of the remaining doors at random (the pick at random part is only significant if there were originally more than 3 doors) and see if that matches the door where the car is.

Finally, we need to run the simulation a thousand times and count how many times each strategy wins:

   +/ (stayWin,switchWin)@scenario"0 i.1000
320 680

Or, we could bundle this all up as a defined word. Here, the (optional) left argument "names" the doors and the right argument says how many simulations to run:

simulate=:3 :0
1 2 3 simulate y
:
pick=. {~ ?@#
scenario=. ((pick@-.,])pick,pick) bind x
stayWin=. =/@}.
switchWin=. pick@(x -. }:) = {:
r=.(stayWin,switchWin)@scenario"0 i.1000
labels=. ];.2 'limit stay switch '
smoutput labels,.":"0 y,+/r
)

Example use:

   simulate 1000
limit 1000
stay 304
switch 696

Or, with more doors (and assuming this does not require new rules about how Monty behavior or how the player behaves):

   1 2 3 4 simulate 1000
limit 1000
stay 233
switch 388

[edit] Java

import java.util.Random;
public class Monty{
public static void main(String[] args){
int switchWins = 0;
int stayWins = 0;
Random gen = new Random();
for(int plays = 0;plays < 32768;plays++ ){
int[] doors = {0,0,0};//0 is a goat, 1 is a car
doors[gen.nextInt(3)] = 1;//put a winner in a random door
int choice = gen.nextInt(3); //pick a door, any door
int shown; //the shown door
do{
shown = gen.nextInt(3);
//don't show the winner or the choice
}while(doors[shown] == 1 || shown == choice);
 
stayWins += doors[choice];//if you won by staying, count it
 
//the switched (last remaining) door is (3 - choice - shown), because 0+1+2=3
switchWins += doors[3 - choice - shown];
}
System.out.println("Switching wins " + switchWins + " times.");
System.out.println("Staying wins " + stayWins + " times.");
}
}

Output:

Switching wins 21924 times.
Staying wins 10844 times.

[edit] JavaScript

[edit] Extensive Solution

This solution can test with n doors, the difference in probability for switching is shown to diminish as the number of doors increases.

 
function montyhall(tests, doors) {
'use strict';
tests = tests ? tests : 1000;
doors = doors ? doors : 3;
var prizeDoor, chosenDoor, shownDoor, switchDoor, chosenWins = 0, switchWins = 0;
 
// randomly pick a door excluding input doors
function pick(excludeA, excludeB) {
var door;
do {
door = Math.floor(Math.random() * doors);
} while (door === excludeA || door === excludeB);
return door;
}
 
// run tests
for (var i = 0; i < tests; i ++) {
 
// pick set of doors
prizeDoor = pick();
chosenDoor = pick();
shownDoor = pick(prizeDoor, chosenDoor);
switchDoor = pick(chosenDoor, shownDoor);
 
// test set for both choices
if (chosenDoor === prizeDoor) {
chosenWins ++;
} else if (switchDoor === prizeDoor) {
switchWins ++;
}
}
 
// results
return {
stayWins: chosenWins + ' ' + (100 * chosenWins / tests) + '%',
switchWins: switchWins + ' ' + (100 * switchWins / tests) + '%'
};
}
 
Output:
 
montyhall(1000, 3)
Object {stayWins: "349 34.9%", switchWins: "651 65.1%"}
montyhall(1000, 4)
Object {stayWins: "253 25.3%", switchWins: "384 38.4%"}
montyhall(1000, 5)
Object {stayWins: "202 20.2%", switchWins: "265 26.5%"}
 

[edit] Basic Solution

 
var totalGames = 10000,
selectDoor = function () {
return Math.floor(Math.random() * 3); // Choose a number from 0, 1 and 2.
},
games = (function () {
var i = 0, games = [];
 
for (; i < totalGames; ++i) {
games.push(selectDoor()); // Pick a door which will hide the prize.
}
 
return games;
}()),
play = function (switchDoor) {
var i = 0, j = games.length, winningDoor, randomGuess, totalTimesWon = 0;
 
for (; i < j; ++i) {
winningDoor = games[i];
randomGuess = selectDoor();
if ((randomGuess === winningDoor && !switchDoor) ||
(randomGuess !== winningDoor && switchDoor))
{
/*
* If I initially guessed the winning door and didn't switch,
* or if I initially guessed a losing door but then switched,
* I've won.
*
* The only time I lose is when I initially guess the winning door
* and then switch.
*/

 
totalTimesWon++;
}
}
return totalTimesWon;
};
 
/*
* Start the simulation
*/

 
console.log("Playing " + totalGames + " games");
console.log("Wins when not switching door", play(false));
console.log("Wins when switching door", play(true));
 
Output:
 
Playing 10000 games
Wins when not switching door 3326
Wins when switching door 6630
 

[edit] Liberty BASIC

 
'adapted from BASIC solution
DIM doors(3) '0 is a goat, 1 is a car
 
total = 10000 'set desired number of iterations
switchWins = 0
stayWins = 0
 
FOR plays = 1 TO total
winner = INT(RND(1) * 3) + 1
doors(winner) = 1'put a winner in a random door
choice = INT(RND(1) * 3) + 1'pick a door, any door
DO
shown = INT(RND(1) * 3) + 1
'don't show the winner or the choice
LOOP WHILE doors(shown) = 1 OR shown = choice
if doors(choice) = 1 then
stayWins = stayWins + 1 'if you won by staying, count it
else
switchWins = switchWins + 1'could have switched to win
end if
doors(winner) = 0 'clear the doors for the next test
NEXT
PRINT "Result for ";total;" games."
PRINT "Switching wins "; switchWins; " times."
PRINT "Staying wins "; stayWins; " times."
 

Output:

Result for 10000 games.
Switching wins 6634 times.
Staying wins 3366 times.

[edit] Lua

function playgame(player)
local car = math.random(3)
local pchoice = player.choice()
local function neither(a, b) --slow, but it works
local el = math.random(3)
return (el ~= a and el ~= b) and el or neither(a, b)
end
local el = neither(car, pchoice)
if(player.switch) then pchoice = neither(pchoice, el) end
player.wins = player.wins + (pchoice == car and 1 or 0)
end
for _, v in ipairs{true, false} do
player = {choice = function() return math.random(3) end,
wins = 0, switch = v}
for i = 1, 20000 do playgame(player) end
print(player.wins)
end

[edit] Mathematica

 montyHall[nGames_] :=
Module[{r, winningDoors, firstChoices, nStayWins, nSwitchWins, s},
r := RandomInteger[{1, 3}, nGames];
winningDoors = r;
firstChoices = r;
nStayWins = Count[Transpose[{winningDoors, firstChoices}], {d_, d_}];
nSwitchWins = nGames - nStayWins;
 
Grid[{{"Strategy", "Wins", "Win %"}, {"Stay", Row[{nStayWins, "/", nGames}], s=N[100 nStayWins/nGames]},
{"Switch", Row[{nSwitchWins, "/", nGames}], 100 - s}}, Frame -> All]]


Usage
montyHall[100000]


MontyHall.jpg

[edit] MATLAB

function montyHall(numDoors,numSimulations)
 
assert(numDoors > 2);
 
function num = randInt(n)
num = floor( n*rand()+1 );
end
 
%The first column will tallie wins, the second losses
switchedDoors = [0 0];
stayed = [0 0];
 
 
for i = (1:numSimulations)
 
availableDoors = (1:numDoors); %Preallocate the available doors
winningDoor = randInt(numDoors); %Define the winning door
playersOriginalChoice = randInt(numDoors); %The player picks his initial choice
 
availableDoors(playersOriginalChoice) = []; %Remove the players choice from the available doors
 
%Pick the door to open from the available doors
openDoor = availableDoors(randperm(numel(availableDoors))); %Sort the available doors randomly
openDoor(openDoor == winningDoor) = []; %Make sure Monty doesn't open the winning door
openDoor = openDoor(randInt(numel(openDoor))); %Choose a random door to open
 
availableDoors(availableDoors==openDoor) = []; %Remove the open door from the available doors
availableDoors(end+1) = playersOriginalChoice; %Put the player's original choice back into the pool of available doors
availableDoors = sort(availableDoors);
 
playersNewChoice = availableDoors(randInt(numel(availableDoors))); %Pick one of the available doors
 
if playersNewChoice == playersOriginalChoice
switch playersNewChoice == winningDoor
case true
stayed(1) = stayed(1) + 1;
case false
stayed(2) = stayed(2) + 1;
otherwise
error 'ERROR'
end
else
switch playersNewChoice == winningDoor
case true
switchedDoors(1) = switchedDoors(1) + 1;
case false
switchedDoors(2) = switchedDoors(2) + 1;
otherwise
error 'ERROR'
end
end
end
 
disp(sprintf('Switch win percentage: %f%%\nStay win percentage: %f%%\n', [switchedDoors(1)/sum(switchedDoors),stayed(1)/sum(stayed)] * 100));
 
end

Output:

>> montyHall(3,100000)
Switch win percentage: 66.705972%
Stay win percentage: 33.420062%


[edit] MAXScript

fn montyHall choice switch =
(
doors = #(false, false, false)
doors[random 1 3] = true
chosen = doors[choice]
if switch then chosen = not chosen
chosen
)
 
fn iterate iterations switched =
(
wins = 0
for i in 1 to iterations do
(
if (montyHall (random 1 3) switched) then
(
wins += 1
)
)
wins * 100 / iterations as float
)
 
iterations = 10000
format ("Stay strategy:%\%\n") (iterate iterations false)
format ("Switch strategy:%\%\n") (iterate iterations true)

Output:

Stay strategy:33.77%
Switch strategy:66.84%

[edit] NetRexx

Translation of: Java
Translation of: REXX
Translation of: PL/I
/* NetRexx ************************************************************
* 30.08.2013 Walter Pachl translated from Java/REXX/PL/I
**********************************************************************/

options replace format comments java crossref savelog symbols nobinary
 
doors = create_doors
switchWins = 0
stayWins = 0
shown=0
Loop plays=1 To 1000000
doors=0
r=r3()
doors[r]=1
choice = r3()
loop Until shown<>choice & doors[shown]=0
shown = r3()
End
If doors[choice]=1 Then
stayWins=stayWins+1
Else
switchWins=switchWins+1
End
Say "Switching wins " switchWins " times."
Say "Staying wins " stayWins " times."
 
method create_doors static returns Rexx
doors = ''
doors[0] = 0
doors[1] = 0
doors[2] = 0
return doors
 
method r3 static
rand=random()
return rand.nextInt(3) + 1

Output

Switching wins  667335  times.
Staying wins    332665  times.   

[edit] Nimrod

Translation of: Python
import math
randomize()
 
proc shuffle[T](x: var seq[T]) =
for i in countdown(x.high, 0):
let j = random(i + 1)
swap(x[i], x[j])
 
# 1 represents a car
# 0 represent a goat
 
var
stay = 0 # amount won if stay in the same position
switch = 0 # amount won if you switch
 
for i in 1..1000:
var lst = @[1,0,0] # one car and two goats
shuffle(lst) # shuffles the list randomly
let ran = random(3) # gets a random number for the random guess
let user = lst[ran] # storing the random guess
del lst, ran # deleting the random guess
 
var huh = 0
for i in lst: # getting a value 0 and deleting it
if i == 0:
del lst, huh # deletes a goat when it finds it
break
inc huh
 
if user == 1: # if the original choice is 1 then stay adds 1
inc stay
 
if lst[0] == 1: # if the switched value is 1 then switch adds 1
inc switch
 
echo "Stay = ",stay
echo "Switch = ",switch

Output:

Stay = 337
Switch = 663

[edit] OCaml

let trials = 10000
 
type door = Car | Goat
 
let play switch =
let n = Random.int 3 in
let d1 = [|Car; Goat; Goat|].(n) in
if not switch then d1
else match d1 with
Car -> Goat
| Goat -> Car
 
let cars n switch =
let total = ref 0 in
for i = 1 to n do
let prize = play switch in
if prize = Car then
incr total
done;
!total
 
let () =
let switch = cars trials true
and stay = cars trials false in
let msg strat n =
Printf.printf "The %s strategy succeeds %f%% of the time.\n"
strat (100. *. (float n /. float trials)) in
msg "switch" switch;
msg "stay" stay

[edit] PARI/GP

test(trials)={
my(stay=0,change=0);
for(i=1,trials,
my(prize=random(3),initial=random(3),opened);
while((opened=random(3))==prize | opened==initial,);
if(prize == initial, stay++, change++)
);
print("Wins when staying: "stay);
print("Wins when changing: "change);
[stay, change]
};
 
test(1e4)

Output:

Wins when staying:  3433
Wins when changing: 6567
%1 = [3433, 6567]

[edit] Perl

#! /usr/bin/perl
use strict;
my $trials = 10000;
 
my $stay = 0;
my $switch = 0;
 
foreach (1 .. $trials)
{
my $prize = int(rand 3);
# let monty randomly choose a door where he puts the prize
my $chosen = int(rand 3);
# let us randomly choose a door...
my $show;
do { $show = int(rand 3) } while $show == $chosen || $show == $prize;
# ^ monty opens a door which is not the one with the
# prize, that he knows it is the one the player chosen
$stay++ if $prize == $chosen;
# ^ if player chose the correct door, player wins only if he stays
$switch++ if $prize == 3 - $chosen - $show;
# ^ if player switches, the door he picks is (3 - $chosen - $show),
# because 0+1+2=3, and he picks the only remaining door that is
# neither $chosen nor $show
}
 
print "Stay win ratio " . (100.0 * $stay/$trials) . "\n";
print "Switch win ratio " . (100.0 * $switch/$trials) . "\n";

[edit] Perl 6

This implementation is parametric over the number of doors. Increasing the number of doors in play makes the superiority of the switch strategy even more obvious.

enum Prize <Car Goat>;
enum Strategy <Stay Switch>;
 
sub play (Strategy $strategy, Int :$doors = 3) returns Prize {
 
# Call the door with a car behind it door 0. Number the
# remaining doors starting from 1.
my Prize @doors = Car, Goat xx $doors - 1;
 
# The player chooses a door.
my Prize $initial_pick = @doors.splice(@doors.keys.pick,1)[0];
 
# Of the n doors remaining, the host chooses n - 1 that have
# goats behind them and opens them, removing them from play.
@doors.splice($_,1)
for pick @doors.elems - 1, grep { @doors[$_] == Goat }, keys @doors;
 
# If the player stays, they get their initial pick. Otherwise,
# they get whatever's behind the remaining door.
return $strategy === Stay ?? $initial_pick !! @doors[0];
}
 
constant TRIALS = 1000;
 
for 3, 10 -> $doors {
my %wins;
say "With $doors doors: ";
for Stay, 'Staying', Switch, 'Switching' -> $s, $name {
for ^TRIALS {
++%wins{$s} if play($s, doors => $doors) == Car;
}
say " $name wins ",
round(100*%wins{$s} / TRIALS),
'% of the time.'
}
}
Output:
With 3 doors: 
  Staying wins 31% of the time.
  Switching wins 68% of the time.
With 10 doors: 
  Staying wins 9% of the time.
  Switching wins 90% of the time.

[edit] PHP

<?php
function montyhall($iterations){
$switch_win = 0;
$stay_win = 0;
 
foreach (range(1, $iterations) as $i){
$doors = array(0, 0, 0);
$doors[array_rand($doors)] = 1;
$choice = array_rand($doors);
do {
$shown = array_rand($doors);
} while($shown == $choice || $doors[$shown] == 1);
 
$stay_win += $doors[$choice];
$switch_win += $doors[3 - $choice - $shown];
}
 
$stay_percentages = ($stay_win/$iterations)*100;
$switch_percentages = ($switch_win/$iterations)*100;
 
echo "Iterations: {$iterations} - ";
echo "Stayed wins: {$stay_win} ({$stay_percentages}%) - ";
echo "Switched wins: {$switch_win} ({$switch_percentages}%)";
}
 
montyhall(10000);
?>

Output:

Iterations: 10000 - Stayed wins: 3331 (33.31%) - Switched wins: 6669 (66.69%)

[edit] PicoLisp

(de montyHall (Keep)
(let (Prize (rand 1 3) Choice (rand 1 3))
(if Keep # Keeping the first choice?
(= Prize Choice) # Yes: Monty's choice doesn't matter
(<> Prize Choice) ) ) ) # Else: Win if your first choice was wrong
 
(prinl
"Strategy KEEP -> "
(let Cnt 0
(do 10000 (and (montyHall T) (inc 'Cnt)))
(format Cnt 2) )
" %" )
 
(prinl
"Strategy SWITCH -> "
(let Cnt 0
(do 10000 (and (montyHall NIL) (inc 'Cnt)))
(format Cnt 2) )
" %" )

Output:

Strategy KEEP    -> 33.01 %
Strategy SWITCH  -> 67.73 %

[edit] PL/I

Translation of: Java
*process source attributes xref;
ziegen: Proc Options(main);
/* REXX ***************************************************************
* 30.08.2013 Walter Pachl derived from Java
**********************************************************************/

Dcl (switchWins,stayWins) Bin Fixed(31) Init(0);
Dcl doors(3) Bin Fixed(31);
Dcl (plays,r,choice) Bin Fixed(31) Init(0);
Dcl c17 Char(17) Init((datetime()));
Dcl p9 Pic'(9)9' def(c17) pos(5);
i=random(p9);
Do plays=1 To 1000000;
doors=0;
r=r3();
doors(r)=1;
choice=r3();
Do Until(shown^=choice & doors(shown)=0);
shown=r3();
End;
If doors(choice)=1 Then
stayWins+=1;
Else
switchWins+=1;
End;
Put Edit("Switching wins ",switchWins," times.")(Skip,a,f(6),a);
Put Edit("Staying wins ",stayWins ," times.")(Skip,a,f(6),a);
 
r3: Procedure Returns(Bin Fixed(31));
/*********************************************************************
* Return a random integer: 1, 2, or 3
*********************************************************************/

Dcl r Bin Float(53);
Dcl res Bin Fixed(31);
r=random();
res=(r*3)+1;
Return(res);
End;
End;

Output:

Switching wins 665908 times.
Staying wins   334092 times.         

[edit] Post Script

Use ghostscript or print this to a postscript printer

%!PS
/Courier  % name the desired font
20 selectfont  % choose the size in points and establish
 % the font as the current one
 
% init random number generator
(%Calendar%) currentdevparams /Second get srand
 
1000000 % iteration count
0 0 % 0 wins on first selection 0 wins on switch
2 index % get iteration count
{
rand 3 mod % winning door
rand 3 mod % first choice
eq {
1 add
}
{
exch 1 add exch
} ifelse
} repeat
 
% compute percentages
2 index div 100 mul exch 2 index div 100 mul
 
 
% display result
70 600 moveto
(Switching the door: ) show
80 string cvs show (%) show
70 700 moveto
(Keeping the same: ) show
80 string cvs show (%) show
 
 
showpage  % print all on the page

Sample output:

Keeping the same: 33.4163%
Switching the door:  66.5837%

[edit] PowerShell

#Declaring variables
$intIterations = 10000
$intKept = 0
$intSwitched = 0
 
#Creating a function
Function Play-MontyHall()
{
#Using a .NET object for randomization
$objRandom = New-Object -TypeName System.Random
 
#Generating the winning door number
$intWin = $objRandom.Next(1,4)
 
#Generating the chosen door
$intChoice = $objRandom.Next(1,4)
 
#Generating the excluded number
#Because there is no method to exclude a number from a range,
#I let it re-generate in case it equals the winning number or
#in case it equals the chosen door.
$intLose = $objRandom.Next(1,4)
While (($intLose -EQ $intWin) -OR ($intLose -EQ $intChoice))
{$intLose = $objRandom.Next(1,4)}
 
#Generating the 'other' door
#Same logic applies as for the chosen door: it cannot be equal
#to the winning door nor to the chosen door.
$intSwitch = $objRandom.Next(1,4)
While (($intSwitch -EQ $intLose) -OR ($intSwitch -EQ $intChoice))
{$intSwitch = $objRandom.Next(1,4)}
 
#Simple counters per win for both categories
#Because a child scope cannot change variables in the parent
#scope, the scope of the counters is expanded script-wide.
If ($intChoice -EQ $intWin)
{$script:intKept++}
If ($intSwitch -EQ $intWin)
{$script:intSwitched++}
 
}
 
#Looping the Monty Hall function for $intIterations times
While ($intIterationCount -LT $intIterations)
{
Play-MontyHall
$intIterationCount++
}
 
#Output
Write-Host "Results through $intIterations iterations:"
Write-Host "Keep  : $intKept ($($intKept/$intIterations*100)%)"
Write-Host "Switch: $intSwitched ($($intSwitched/$intIterations*100)%)"
Write-Host ""

Output:

Results through 10000 iterations:
Keep  : 3336 (33.36%)
Switch: 6664 (66.64%)

[edit] PureBasic

Structure wins
stay.i
redecide.i
EndStructure
 
#goat = 0
#car = 1
Procedure MontyHall(*results.wins)
Dim Doors(2)
Doors(Random(2)) = #car
 
player = Random(2)
Select Doors(player)
Case #car
*results\redecide + #goat
*results\stay + #car
Case #goat
*results\redecide + #car
*results\stay + #goat
EndSelect
EndProcedure
 
OpenConsole()
#Tries = 1000000
 
Define results.wins
 
For i = 1 To #Tries
MontyHall(@results)
Next
 
PrintN("Trial runs for each option: " + Str(#Tries))
PrintN("Wins when redeciding: " + Str(results\redecide) + " (" + StrD(results\redecide / #Tries * 100, 2) + "% chance)")
PrintN("Wins when sticking: " + Str(results\stay) + " (" + StrD(results\stay / #Tries * 100, 2) + "% chance)")
Input()
Output:
Trial runs for each option: 1000000
Wins when redeciding: 666459 (66.65% chance)
Wins when sticking:   333541 (33.35% chance)

[edit] Python

'''
I could understand the explanation of the Monty Hall problem
but needed some more evidence
 
References:
http://www.bbc.co.uk/dna/h2g2/A1054306
http://en.wikipedia.org/wiki/Monty_Hall_problem especially:
http://en.wikipedia.org/wiki/Monty_Hall_problem#Increasing_the_number_of_doors
'''

from random import randrange
 
doors, iterations = 3,100000 # could try 100,1000
 
def monty_hall(choice, switch=False, doorCount=doors):
# Set up doors
door = [False]*doorCount
# One door with prize
door[randrange(doorCount)] = True
 
chosen = door[choice]
 
unpicked = door
del unpicked[choice]
 
# Out of those unpicked, the alternative is either:
# the prize door, or
# an empty door if the initial choice is actually the prize.
alternative = True in unpicked
 
if switch:
return alternative
else:
return chosen
 
print "\nMonty Hall problem simulation:"
print doors, "doors,", iterations, "iterations.\n"
 
print "Not switching allows you to win",
print sum(monty_hall(randrange(3), switch=False)
for x in range(iterations)),
print "out of", iterations, "times."
print "Switching allows you to win",
print sum(monty_hall(randrange(3), switch=True)
for x in range(iterations)),
print "out of", iterations, "times.\n"

Sample output:

Monty Hall problem simulation:
3 doors, 100000 iterations.

Not switching allows you to win 33337 out of 100000 times.
Switching allows you to win 66529 out of 100000 times.


[edit] Python 3 version:

Another (simpler in my opinion), way to do this is below, also in python 3:

import random
#1 represents a car
#0 represent a goat
 
stay = 0 #amount won if stay in the same position
switch = 0 # amount won if you switch
 
for i in range(1000):
lst = [1,0,0] # one car and two goats
random.shuffle(lst) # shuffles the list randomly
 
ran = random.randrange(3) # gets a random number for the random guess
 
user = lst[ran] #storing the random guess
 
del(lst[ran]) # deleting the random guess
 
huh = 0
for i in lst: # getting a value 0 and deleting it
if i ==0:
del(lst[huh]) # deletes a goat when it finds it
break
huh+=1
 
if user ==1: # if the original choice is 1 then stay adds 1
stay+=1
 
if lst[0] == 1: # if the switched value is 1 then switch adds 1
switch+=1
 
print("Stay =",stay)
print("Switch = ",switch)
#Done by Sam Witton 09/04/2014

[edit] R

# Since R is a vector based language that penalizes for loops, we will avoid
# for-loops, instead using "apply" statement variants (like "map" in other
# functional languages).
 
set.seed(19771025) # set the seed to set the same results as this code
N <- 10000 # trials
true_answers <- sample(1:3, N, replace=TRUE)
 
# We can assme that the contestant always choose door 1 without any loss of
# generality, by equivalence. That is, we can always relabel the doors
# to make the user-chosen door into door 1.
# Thus, the host opens door '2' unless door 2 has the prize, in which case
# the host opens door 3.
 
host_opens <- 2 + (true_answers == 2)
other_door <- 2 + (true_answers != 2)
 
## if always switch
summary( other_door == true_answers )
## if we never switch
summary( true_answers == 1)
## if we randomly switch
random_switch <- other_door
random_switch[runif(N) >= .5] <- 1
summary(random_switch == true_answers)
 
 
 
## To go with the exact parameters of the Rosetta challenge, complicating matters....
## Note that the player may initially choose any of the three doors (not just Door 1),
## that the host opens a different door revealing a goat (not necessarily Door 3), and
## that he gives the player a second choice between the two remaining unopened doors.
 
N <- 10000 #trials
true_answers <- sample(1:3, N, replace=TRUE)
user_choice <- sample(1:3, N, replace=TRUE)
## the host_choice is more complicated
host_chooser <- function(user_prize) {
# this could be cleaner
bad_choices <- unique(user_prize)
# in R, the x[-vector] form implies, choose the indices in x not in vector
choices <- c(1:3)[-bad_choices]
# if the first arg to sample is an int, it treats it as the number of choices
if (length(choices) == 1) { return(choices)}
else { return(sample(choices,1))}
}
 
host_choice <- apply( X=cbind(true_answers,user_choice), FUN=host_chooser,MARGIN=1)
not_door <- function(x){ return( (1:3)[-x]) } # we could also define this
# directly at the FUN argument following
other_door <- apply( X = cbind(user_choice,host_choice), FUN=not_door, MARGIN=1)
 
 
## if always switch
summary( other_door == true_answers )
## if we never switch
summary( true_answers == user_choice)
## if we randomly switch
random_switch <- user_choice
change <- runif(N) >= .5
random_switch[change] <- other_door[change]
summary(random_switch == true_answers)


Results: 

> ## if always switch
> summary( other_door == true_answers )
   Mode   FALSE    TRUE 
logical    3298    6702 
> ## if we never switch
> summary( true_answers == 1)
   Mode   FALSE    TRUE 
logical    6702    3298 
> ## if we randomly switch
> summary(random_switch == true_answers)
   Mode   FALSE    TRUE 
logical    5028    4972 


> ## if always switch
> summary( other_door == true_answers )
   Mode   FALSE    TRUE 
logical    3295    6705 
> ## if we never switch
> summary( true_answers == user_choice)
   Mode   FALSE    TRUE 
logical    6705    3295 
> ## if we randomly switch
> summary(random_switch == true_answers)
   Mode   FALSE    TRUE 
logical    4986    5014 
# As above, but generalized to K number of doors

K = 4     # number of doors
N = 1e4   # number of simulation trials

chooser <- function(x) { i <- (1:K)[-x]; if (length(i)>1) sample(i,1) else i }

p100 <- function(...) { cat("\nNumber of doors:", K,
       "\nSimulation yields % winning probability:",
       " (2nd choice after host reveal)\n");
        print(c(...) * 100, digits=3) }

prize_door <- sample(1:K, N, replace=TRUE)
first_choice <- sample(1:K, N, replace=TRUE)

host_opens <- apply(cbind(prize_door, first_choice), 1, chooser)
second_choice <- apply(cbind(host_opens, first_choice), 1, chooser)

p100("By first choice" = (Pr.first_win <- mean(first_choice == prize_door)),
     "By second choice" = (Pr.second_win <- mean(second_choice == prize_door)),
     "  Change gain" = Pr.second_win / Pr.first_win - 1)

#-------
#
# Sample output:

Number of doors: 4 
Simulation yields % winning probability:  (2nd choice after host reveal)
 By first choice By second choice      Change gain 
            24.7             36.5             48.0 

[edit] Racket

 
#lang racket
 
(define (get-last-door a b) ; assumes a != b
(vector-ref '#(- 2 1
2 - 0
1 0 -)
(+ a (* 3 b))))
 
(define (run-game strategy)
(define car-door (random 3))
(define first-choice (random 3))
(define revealed-goat
(if (= car-door first-choice)
(let ([r (random 2)]) (if (<= car-door r) (add1 r) r)) ; random
(get-last-door car-door first-choice))) ; reveal goat
(define final-choice (strategy first-choice revealed-goat))
(define win? (eq? final-choice car-door))
 ;; (printf "car: ~s\nfirst: ~s\nreveal: ~s\nfinal: ~s\n => ~s\n\n"
 ;; car-door first-choice revealed-goat final-choice
 ;; (if win? 'win 'lose))
win?)
 
(define (keep-choice first-choice revealed-goat)
first-choice)
 
(define (change-choice first-choice revealed-goat)
(get-last-door first-choice revealed-goat))
 
(define (test-strategy strategy)
(define N 10000000)
(define wins (for/sum ([i (in-range N)]) (if (run-game strategy) 1 0)))
(printf "~a: ~a%\n"
(object-name strategy)
(exact->inexact (/ wins N 1/100))))
 
(for-each test-strategy (list keep-choice change-choice))
 

Sample Output:

keep-choice: 33.33054%
change-choice: 66.67613%

[edit] REXX

[edit] version 1

Translation of: Java
/* REXX ***************************************************************
* 30.08.2013 Walter Pachl derived from Java
**********************************************************************/

Call time 'R'
switchWins = 0;
stayWins = 0
Do plays = 1 To 1000000
doors.=0
r=r3()
doors.r=1
choice = r3()
Do Until shown<>choice & doors.shown=0
shown = r3()
End
If doors.choice=1 Then
stayWins=stayWins+1
Else
switchWins=switchWins+1
End
Say "Switching wins " switchWins " times."
Say "Staying wins " stayWins " times."
Say 'REXX:' time('E') 'seconds'
Call time 'R'
'ziegen'
Say 'PL/I:' time('E') 'seconds'
Say ' '
Call time 'R'
'java ziegen'
Say 'NetRexx:' time('E') 'seconds'
Exit
r3: Return random(2)+1

Output for 1000000 samples:

Switching wins  666442  times.
Staying wins    333558  times.
REXX:   4.321000 seconds

Switching wins 665908 times.
Staying wins   334092 times.
PL/I:   0.328000 seconds

Switching wins  667335  times.
Staying wins    332665  times.
NetRexx: 2.042000 seconds        

[edit] version 2

/*REXX program simulates a # of trials of the classic Monty Hall problem*/
parse arg t .; if t=='' then t=1000000 /*Not specified? Then use default*/
wins.=0 /*wins.0=stay; wins.1=switching.*/
/*door values: 0=goat 1=car */
do t /*perform this loop T times. */
door.=0 /*set all doors to zero. */
car=random(1,3); door.car=1 /*TV show hides a car randomly. */
 ?=random(1,3)  ; _=door.? /*contestant picks a random door.*/
wins._=wins._+1 /*bump the type of win strategy. */
end /*DO t*/
 
say 'switching wins ' format(wins.0/t*100,,1)"% of the time."
say ' staying wins ' format(wins.1/t*100,,1)"% of the time."; say
say 'performed' t "times." /*stick a fork in it, we're done.*

output when using the default number of trials   (one million):

switching wins  66.7%  of the time.
  staying wins  33.3%  of the time.

performed 1000000 times.

[edit] Ruby

n = 10_000                  #number of times to play
 
stay = switch = 0 #sum of each strategy's wins
 
n.times do #play the game n times
 
#the doors reveal 2 goats and a car
doors = [ :goat, :goat, :car ].shuffle
 
#random guess
guess = rand(3)
 
#random door shown, but it is neither the guess nor the car
begin shown = rand(3) end while shown == guess || doors[shown] == :car
 
if doors[guess] == :car
#staying with the initial guess wins if the initial guess is the car
stay += 1
else
#switching guesses wins if the unshown door is the car
switch += 1
end
 
end
 
puts "Staying wins %.2f%% of the time." % (100.0 * stay / n)
puts "Switching wins %.2f%% of the time." % (100.0 * switch / n)

Sample Output:

Staying wins 33.84% of the time.
Switching wins 66.16% of the time.

[edit] Run BASIC

' adapted from BASIC solution
 
input "Number of tries;";tries ' gimme the number of iterations
FOR plays = 1 TO tries
winner = INT(RND(1) * 3) + 1
doors(winner) = 1 'put a winner in a random door
choice = INT(RND(1) * 3) + 1 'pick a door please
[DO] shown = INT(RND(1) * 3) + 1
' ------------------------------------------
' don't show the winner or the choice
if doors(shown) = 1 then goto [DO]
if shown = choice then goto [DO]
if doors(choice) = 1 then
stayWins = stayWins + 1 ' if you won by staying, count it
else
switchWins = switchWins + 1 ' could have switched to win
end if
doors(winner) = 0 'clear the doors for the next test
NEXT
PRINT " Result for ";tries;" games."
PRINT "Switching wins ";switchWins; " times."
PRINT " Staying wins ";stayWins; " times."

[edit] Scala

import scala.util.Random
 
object MontyHallSimulation {
def main(args: Array[String]) {
val samples = if (args.size == 1 && (args(0) matches "\\d+")) args(0).toInt else 1000
val doors = Set(0, 1, 2)
var stayStrategyWins = 0
var switchStrategyWins = 0
 
1 to samples foreach { _ =>
val prizeDoor = Random shuffle doors head;
val choosenDoor = Random shuffle doors head;
val hostDoor = Random shuffle (doors - choosenDoor - prizeDoor) head;
val switchDoor = doors - choosenDoor - hostDoor head;
 
(choosenDoor, switchDoor) match {
case (`prizeDoor`, _) => stayStrategyWins += 1
case (_, `prizeDoor`) => switchStrategyWins += 1
}
}
 
def percent(n: Int) = n * 100 / samples
 
val report = """|%d simulations were ran.
|Staying won %d times (%d %%)
|Switching won %d times (%d %%)"
"".stripMargin
 
println(report
format (samples,
stayStrategyWins, percent(stayStrategyWins),
switchStrategyWins, percent(switchStrategyWins)))
}
}

Sample:

1000 simulations were ran.
Staying won 333 times (33 %)
Switching won 667 times (66 %)

[edit] Scheme

(define (random-from-list list) (list-ref list (random (length list))))
(define (random-permutation list)
(if (null? list)
'()
(let* ((car (random-from-list list))
(cdr (random-permutation (remove car list))))
(cons car cdr))))
(define (random-configuration) (random-permutation '(goat goat car)))
(define (random-door) (random-from-list '(0 1 2)))
 
(define (trial strategy)
(define (door-with-goat-other-than door strategy)
(cond ((and (not (= 0 door)) (equal? (list-ref strategy 0) 'goat)) 0)
((and (not (= 1 door)) (equal? (list-ref strategy 1) 'goat)) 1)
((and (not (= 2 door)) (equal? (list-ref strategy 2) 'goat)) 2)))
(let* ((configuration (random-configuration))
(players-first-guess (strategy `(would-you-please-pick-a-door?)))
(door-to-show-player (door-with-goat-other-than players-first-guess
configuration))
(players-final-guess (strategy `(there-is-a-goat-at/would-you-like-to-move?
,players-first-guess
,door-to-show-player))))
(if (equal? (list-ref configuration players-final-guess) 'car)
'you-win!
'you-lost)))
 
(define (stay-strategy message)
(case (car message)
((would-you-please-pick-a-door?) (random-door))
((there-is-a-goat-at/would-you-like-to-move?)
(let ((first-choice (cadr message)))
first-choice))))
 
(define (switch-strategy message)
(case (car message)
((would-you-please-pick-a-door?) (random-door))
((there-is-a-goat-at/would-you-like-to-move?)
(let ((first-choice (cadr message))
(shown-goat (caddr message)))
(car (remove first-choice (remove shown-goat '(0 1 2))))))))
 
(define-syntax repeat
(syntax-rules ()
((repeat <n> <body> ...)
(let loop ((i <n>))
(if (zero? i)
'()
(cons ((lambda () <body> ...))
(loop (- i 1))))))))
 
(define (count element list)
(if (null? list)
0
(if (equal? element (car list))
(+ 1 (count element (cdr list)))
(count element (cdr list)))))
 
(define (prepare-result strategy results)
`(,strategy won with probability
,(exact->inexact (* 100 (/ (count 'you-win! results) (length results)))) %))
 
(define (compare-strategies times)
(append
(prepare-result 'stay-strategy (repeat times (trial stay-strategy)))
'(and)
(prepare-result 'switch-strategy (repeat times (trial switch-strategy)))))
 
;; > (compare-strategies 1000000)
;; (stay-strategy won with probability 33.3638 %
;; and switch-strategy won with probability 66.716 %)

[edit] Scilab

//  How it works:
// MontyHall() is a function with argument switch:
// it will be called 100000 times with switch=%T,
// and another 100000 times with switch=%F
 
function win=MontyHall(switch) //If switch==%T the player will switch
doors=zeros(1,3) //All goats
car=grand(1,1,'uin',1,3)
a(car)=1 //Place a car somewher
pick=grand(1,1,'uin',1,3) //The player picks...
if pick==car then //If the player picks right...
if switch==%T then //...and switches he will be wrong
win=%F
else //...but if he doesn't, he will be right
win=%T
end
else //If the player picks a goat...
if switch==%T then //...and switches: the other door with the goat shall be
win=%T // opened: the player will switch to the car and win
else //...but if he doesn't, he will remain by his goat
win=%F
end
end
endfunction
 
wins_switch=0
wins_stay=0
games=100000
for i=1:games
if MontyHall(%T)==%T then
wins_switch=wins_switch+1
end
if MontyHall(%F)==%T then
wins_stay=wins_stay+1
end
end
disp("Switching, one wins"+ascii(10)+string(wins_switch)+" games out of "+string(games))
disp("Staying, one wins"+ascii(10)+string(wins_stay)+" games out of "+string(games))

Output:

 Switching, one wins
 66649 games out of 100000   
 
 Staying, one wins
 33403 games out of 100000  

[edit] Seed7

$ include "seed7_05.s7i";
 
const proc: main is func
local
var integer: switchWins is 0;
var integer: stayWins is 0;
var integer: winner is 0;
var integer: choice is 0;
var integer: shown is 0;
var integer: plays is 0;
begin
for plays range 1 to 10000 do
winner := rand(1, 3);
choice := rand(1, 3);
repeat
shown := rand(1, 3)
until shown <> winner and shown <> choice;
stayWins +:= ord(choice = winner);
switchWins +:= ord(6 - choice - shown = winner);
end for;
writeln("Switching wins " <& switchWins <& " times");
writeln("Staying wins " <& stayWins <& " times");
end func;

Output:

Switching wins 6654 times
Staying wins 3346 times

[edit] Tcl

A simple way of dealing with this one, based on knowledge of the underlying probabilistic system, is to use code like this:

set stay 0; set change 0; set total 10000
for {set i 0} {$i<$total} {incr i} {
if {int(rand()*3) == int(rand()*3)} {
incr stay
} else {
incr change
}
}
puts "Estimate: $stay/$total wins for staying strategy"
puts "Estimate: $change/$total wins for changing strategy"

But that's not really the point of this challenge; it should add the concealing factors too so that we're simulating not just the solution to the game, but also the game itself. (Note that we are using Tcl's lists here to simulate sets.)

We include a third strategy that is proposed by some people (who haven't thought much about it) for this game: just picking at random between all the doors offered by Monty the second time round.

package require Tcl 8.5
 
# Utility: pick a random item from a list
proc pick list {
lindex $list [expr {int(rand()*[llength $list])}]
}
# Utility: remove an item from a list if it is there
proc remove {list item} {
set idx [lsearch -exact $list $item]
return [lreplace $list $idx $idx]
}
 
# Codify how Monty will present the new set of doors to choose between
proc MontyHallAction {doors car picked} {
set unpicked [remove $doors $picked]
if {$car in $unpicked} {
# Remove a random unpicked door without the car behind it
set carless [remove $unpicked $car]
return [list {*}[remove $carless [pick $carless]] $car]
# Expressed this way so Monty Hall isn't theoretically
# restricted to using 3 doors, though that could be written
# as just: return [list $car]
} else {
# Monty has a real choice now...
return [remove $unpicked [pick $unpicked]]
}
}
 
# The different strategies you might choose
proc Strategy:Stay {originalPick otherChoices} {
return $originalPick
}
proc Strategy:Change {originalPick otherChoices} {
return [pick $otherChoices]
}
proc Strategy:PickAnew {originalPick otherChoices} {
return [pick [list $originalPick {*}$otherChoices]]
}
 
# Codify one round of the game
proc MontyHallGameRound {doors strategy winCounter} {
upvar 1 $winCounter wins
set car [pick $doors]
set picked [pick $doors]
set newDoors [MontyHallAction $doors $car $picked]
set picked [$strategy $picked $newDoors]
# Check for win...
if {$car eq $picked} {
incr wins
}
}
 
# We're always using three doors
set threeDoors {a b c}
set stay 0; set change 0; set anew 0
set total 10000
# Simulate each of the different strategies
for {set i 0} {$i<$total} {incr i} {
MontyHallGameRound $threeDoors Strategy:Stay stay
MontyHallGameRound $threeDoors Strategy:Change change
MontyHallGameRound $threeDoors Strategy:PickAnew anew
}
# Print the results
puts "Estimate: $stay/$total wins for 'staying' strategy"
puts "Estimate: $change/$total wins for 'changing' strategy"
puts "Estimate: $anew/$total wins for 'picking anew' strategy"

This might then produce output like

Estimate: 3340/10000 wins for 'staying' strategy
Estimate: 6733/10000 wins for 'changing' strategy
Estimate: 4960/10000 wins for 'picking anew' strategy

Of course, this challenge could also be tackled by putting up a GUI and letting the user be the source of the randomness. But that's moving away from the letter of the challenge and takes a lot of effort anyway...

[edit] UNIX Shell

Works with: 2.x version and most bash-compatible unix shells
#!/bin/bash
# Simulates the "monty hall" probability paradox and shows results.
# http://en.wikipedia.org/wiki/Monty_Hall_problem
# (should rewrite this in C for faster calculating of huge number of rounds)
# (Hacked up by Éric Tremblay, 07.dec.2010)
 
num_rounds=10 #default number of rounds
num_doors=3 # default number of doors
[ "$1" = "" ] || num_rounds=$[$1+0]
[ "$2" = "" ] || num_doors=$[$2+0]
 
nbase=1 # or 0 if we want to see door numbers zero-based
num_win=0; num_lose=0
 
echo "Playing $num_rounds times, with $num_doors doors."
[ "$num_doors" -lt 3 ] && {
echo "Hey, there has to be at least 3 doors!!"
exit 1
}
echo
 
function one_round() {
winning_door=$[$RANDOM % $num_doors ]
player_picks_door=$[$RANDOM % $num_doors ]
 
# Host leaves this door AND the player's first choice closed, opens all others
# (this WILL loop forever if there is only 1 door)
host_skips_door=$winning_door
while [ "$host_skips_door" = "$player_picks_door" ]; do
#echo -n "(Host looks at door $host_skips_door...) "
host_skips_door=$[$RANDOM % $num_doors]
done
 
# Output the result of this round
#echo "Round $[$nbase+current_round]: "
echo -n "Player chooses #$[$nbase+$player_picks_door]. "
[ "$num_doors" -ge 10 ] &&
# listing too many door numbers (10 or more) will just clutter the output
echo -n "Host opens all except #$[$nbase+$host_skips_door] and #$[$nbase+$player_picks_door]. " \
|| {
# less than 10 doors, we list them one by one instead of "all except ?? and ??"
echo -n "Host opens"
host_opens=0
while [ "$host_opens" -lt "$num_doors" ]; do
[ "$host_opens" != "$host_skips_door" ] && [ "$host_opens" != "$player_picks_door" ] && \
echo -n " #$[$nbase+$host_opens]"
host_opens=$[$host_opens+1]
done
echo -n " "
}
echo -n "(prize is behind #$[$nbase+$winning_door]) "
echo -n "Switch from $[$nbase+$player_picks_door] to $[$nbase+$host_skips_door]: "
[ "$winning_door" = "$host_skips_door" ] && {
echo "WIN."
num_win=$[num_win+1]
} || {
echo "LOSE."
num_lose=$[num_lose+1]
}
} # end of function one_round
 
# ok, let's go
current_round=0
while [ "$num_rounds" -gt "$current_round" ]; do
one_round
current_round=$[$current_round+1]
done
 
echo
echo "Wins (switch to remaining door): $num_win"
echo "Losses (first guess was correct): $num_lose"
exit 0

Output of a few runs:

$ ./monty_hall_problem.sh
Playing 10 times, with 3 doors.

Player chooses #2. Host opens #3 (prize is behind #1) Switch from 2 to 1: WIN.
Player chooses #1. Host opens #3 (prize is behind #2) Switch from 1 to 2: WIN.
Player chooses #2. Host opens #3 (prize is behind #2) Switch from 2 to 1: LOSE.
Player chooses #1. Host opens #2 (prize is behind #1) Switch from 1 to 3: LOSE.
Player chooses #2. Host opens #3 (prize is behind #1) Switch from 2 to 1: WIN.
Player chooses #2. Host opens #1 (prize is behind #2) Switch from 2 to 3: LOSE.
Player chooses #3. Host opens #1 (prize is behind #2) Switch from 3 to 2: WIN.
Player chooses #2. Host opens #1 (prize is behind #3) Switch from 2 to 3: WIN.
Player chooses #1. Host opens #3 (prize is behind #1) Switch from 1 to 2: LOSE.
Player chooses #1. Host opens #2 (prize is behind #3) Switch from 1 to 3: WIN.

Wins (switch to remaining door):  6
Losses (first guess was correct): 4


$ ./monty_hall_problem.sh 5 10
Playing 5 times, with 10 doors.

Player chooses #1. Host opens all except #10 and #1. (prize is behind #10) Switch from 1 to 10: WIN.
Player chooses #7. Host opens all except #8 and #7. (prize is behind #8) Switch from 7 to 8: WIN.
Player chooses #6. Host opens all except #1 and #6. (prize is behind #1) Switch from 6 to 1: WIN.
Player chooses #8. Host opens all except #3 and #8. (prize is behind #8) Switch from 8 to 3: LOSE.
Player chooses #6. Host opens all except #5 and #6. (prize is behind #5) Switch from 6 to 5: WIN.

Wins (switch to remaining door):  4
Losses (first guess was correct): 1


$ ./monty_hall_problem.sh 1000
Playing 1000 times, with 3 doors.

Player chooses #2. Host opens #1 (prize is behind #2) Switch from 2 to 3: LOSE.
Player chooses #3. Host opens #1 (prize is behind #2) Switch from 3 to 2: WIN.
[ ... ]
Player chooses #1. Host opens #3 (prize is behind #2) Switch from 1 to 2: WIN.
Player chooses #3. Host opens #2 (prize is behind #1) Switch from 3 to 1: WIN.

Wins (switch to remaining door):  655
Losses (first guess was correct): 345

[edit] Ursala

This is the same algorithm as the Perl solution. Generate two lists of 10000 uniformly distributed samples from {1,2,3}, count each match as a win for the staying strategy, and count each non-match as a win for the switching strategy.

#import std
#import nat
#import flo
 
rounds = 10000
 
car_locations = arc{1,2,3}* iota rounds
initial_choices = arc{1,2,3}* iota rounds
 
staying_wins = length (filter ==) zip(car_locations,initial_choices)
switching_wins = length (filter ~=) zip(car_locations,initial_choices)
 
format = printf/'%0.2f'+ (times\100.+ div+ float~~)\rounds
 
#show+
 
main = ~&plrTS/<'stay: ','switch: '> format* <staying_wins,switching_wins>

Output will vary slightly for each run due to randomness.

stay:   33.95
switch: 66.05

[edit] Vedit macro language

Translation of: BASIC

Vedit macro language does not have random number generator, so one is implemented in subroutine RANDOM (the algorithm was taken from ANSI C library).

#90 = Time_Tick			// seed for random number generator
#91 = 3 // random numbers in range 0 to 2
#1 = 0 // wins for "always stay" strategy
#2 = 0 // wins for "always switch" strategy
for (#10 = 0; #10 < 10000; #10++) { // 10,000 iterations
Call("RANDOM")
#3 = Return_Value // #3 = winning door
Call("RANDOM")
#4 = Return_Value // #4 = players choice
do {
Call("RANDOM")
#5 = Return_Value // #5 = door to open
} while (#5 == #3 || #5 == #4)
if (#3 == #4) { // original choice was correct
#1++
}
if (#3 == 3 - #4 - #5) { // switched choice was correct
#2++
}
}
Ins_Text("Staying wins: ") Num_Ins(#1)
Ins_Text("Switching wins: ") Num_Ins(#2)
return
 
//--------------------------------------------------------------
// Generate random numbers in range 0 <= Return_Value < #91
// #90 = Seed (0 to 0x7fffffff)
// #91 = Scaling (0 to 0xffff)
 
:RANDOM:
#92 = 0x7fffffff / 48271
#93 = 0x7fffffff % 48271
#90 = (48271 * (#90 % #92) - #93 * (#90 / #92)) & 0x7fffffff
return ((#90 & 0xffff) * #91 / 0x10000)

Sample output:

Staying winns:    3354
Switching winns:  6646

[edit] XPL0

def Games = 10000;              \number of games simulated
int Game, Wins;
include c:\cxpl\codes;
 
proc Play(Switch); \Play one game
int Switch;
int Car, Player, Player0, Monty;
[Car:= Ran(3); \randomly place car behind a door
Player0:= Ran(3); \player randomly chooses a door
repeat Monty:= Ran(3); \Monty opens door revealing a goat
until Monty # Car and Monty # Player0;
if Switch then \player switches to remaining door
repeat Player:= Ran(3);
until Player # Player0 and Player # Monty
else Player:= Player0; \player sticks with original door
if Player = Car then Wins:= Wins+1;
];
 
[Format(2,1);
Text(0, "Not switching doors wins car in ");
Wins:= 0;
for Game:= 0 to Games-1 do Play(false);
RlOut(0, float(Wins)/float(Games)*100.0);
Text(0, "% of games.^M^J");
 
Text(0, "But switching doors wins car in ");
Wins:= 0;
for Game:= 0 to Games-1 do Play(true);
RlOut(0, float(Wins)/float(Games)*100.0);
Text(0, "% of games.^M^J");
]

Example output:

Not switching doors wins car in 33.7% of games.
But switching doors wins car in 66.7% of games.

[edit] zkl

Translation of: Go
const games=0d100_000;
 
reg switcherWins=0, keeperWins=0, shown=0;
do(games){
doors := L(0,0,0);
doors[(0).random(3)] = 1; // Set which one has the car
choice := (0).random(3); // Choose a door
while(1){ shown = (0).random(3);
if (not (shown == choice or doors[shown] == 1)) break; }
switcherWins += doors[3 - choice - shown];
keeperWins += doors[choice];
}
 
"Switcher Wins: %,d (%3.2f%%)".fmt(
switcherWins, switcherWins.toFloat() / games * 100).println();
"Keeper Wins: %,d (%3.2f%%)".fmt(
keeperWins, keeperWins.toFloat() / games * 100).println();
Output:
Switcher Wins: 66,730 (66.73%)
Keeper Wins: 33,270 (33.27%)
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