Monty Hall Problem

Question

Through trying to explain the Monty Hall problem to a friend during class yesterday, we ended up coding it in Python to prove that if you always swap, you will win 2/3 times. We came up with this:

import random as r

#iterations = int(raw_input("How many iterations? >> "))
iterations = 100000

doors = ["goat", "goat", "car"]
wins = 0.0
looses = 0.0

for i in range(iterations):
 n = r.randrange(0,3)

 choice = doors[n]
 if n == 0:
  #print "You chose door 1."
  #print "Monty opens door 2. There is a goat behind this door."
  #print "You swapped to door 3."
  wins += 1
  #print "You won a " + doors[2] + "\n"
 elif n == 1:
  #print "You chose door 2."
  #print "Monty opens door 1. There is a goat behind this door."
  #print "You swapped to door 3."
  wins += 1
  #print "You won a " + doors[2] + "\n"
 elif n == 2:
  #print "You chose door 3."
  #print "Monty opens door 2. There is a goat behind this door."
  #print "You swapped to door 1."
  looses += 1
  #print "You won a " + doors[0] + "\n"
 else:
  print "You screwed up"

percentage = (wins/iterations) * 100
print "Wins: " + str(wins)
print "Looses: " + str(looses)
print "You won " + str(percentage) + "% of the time"

My friend thought this was a good way of going about it (and is a good simulation for it), but I have my doubts and concerns. Is it actually random enough?

The problem I have with it is that the all choices are kind of hard coded in.

Is this a good or bad 'simulation' for the Monty Hall problem? How come?

Can you come up with a better version?

Answer 1

Monty never opens the door with the car - that's the whole point of the show (he isn't your friend an has knowledge of what is behind each door)

Answer 2

You mentioned that all the choices are hardcoded in. But if you look closer, you'll notice that what you think are 'choices' are actually not choices at all. Monty's decision is without loss of generality since he always chooses the door with the goat behind it. Your swapping is always determined by what Monty chooses, and since Monty's "choice" was actually not a choice, neither is yours. Your simulation gives the correct results..

Answer 3

Your solution is fine, but if you want a stricter simulation of the problem as posed (and somewhat higher-quality Python;-), try:

import random

iterations = 100000

doors = ["goat"] * 2 + ["car"]
change_wins = 0
change_loses = 0

for i in xrange(iterations):
    random.shuffle(doors)
    # you pick door n:
    n = random.randrange(3)
    # monty picks door k, k!=n and doors[k]!="car"
    sequence = range(3)
    random.shuffle(sequence)
    for k in sequence:
        if k == n or doors[k] == "car":
            continue
    # now if you change, you lose iff doors[n]=="car"
    if doors[n] == "car":
        change_loses += 1
    else:
        change_wins += 1

print "Changing has %s wins and %s losses" % (change_wins, change_loses)
perc = (100.0 * change_wins) / (change_wins + change_loses)
print "IOW, by changing you win %.1f%% of the time" % perc

a typical output is:

Changing has 66721 wins and 33279 losses
IOW, by changing you win 66.7% of the time

Answer 4

I like something like this.

#!/usr/bin/python                                                                                                            
import random
CAR   = 1
GOAT  = 0

def one_trial( doors, switch=False ):
    """One trial of the Monty Hall contest."""

    random.shuffle( doors )
    first_choice = doors.pop( )
    if switch==False:
        return first_choice
    elif doors.__contains__(CAR):
        return CAR
    else:
        return GOAT


def n_trials( switch=False, n=10 ):
    """Play the game N times and return some stats."""
    wins = 0
    for n in xrange(n):
        doors = [CAR, GOAT, GOAT]
        wins += one_trial( doors, switch=switch )

    print "won:", wins, "lost:", (n-wins), "avg:", (float(wins)/float(n))


if __name__=="__main__":
    import sys
    n_trials( switch=eval(sys.argv[1]), n=int(sys.argv[2]) )

$ ./montyhall.py True 10000
won: 6744 lost: 3255 avg: 0.674467446745

Answer 5

Here is an interactive version:

from random import shuffle, choice
cars,goats,iters= 0, 0, 100
for i in range(iters):
    doors = ['goat A', 'goat B', 'car']
    shuffle(doors)
    moderator_door = 'car'
    #Turn 1:
    selected_door = choice(doors)
    print selected_door
    doors.remove(selected_door)
    print 'You have selected a door with an unknown object'
    #Turn 2:
    while moderator_door == 'car':
        moderator_door = choice(doors)
    doors.remove(moderator_door)
    print 'Moderator has opened a door with ', moderator_door
    #Turn 3:
    decision=raw_input('Wanna change your door? [yn]')
    if decision=='y':
        prise = doors[0]
        print 'You have a door with ', prise
    elif decision=='n':
        prise = selected_door
        print 'You have a door with ', prise
    else:
        prise = 'ERROR'
        iters += 1
        print 'ERROR:unknown command'
    if prise == 'car':
        cars += 1
    elif prise != 'ERROR':
        goats += 1
print '==============================='
print '          RESULTS              '
print '==============================='
print 'Goats:', goats
print 'Cars :', cars

Answer 6

I hadn't heard of the Monty Hall Problem before I stumbled across this question. I thought it was interesting, so I read about it and created a c# simulation. It's kind of goofy since it simulates the game-show and not just the problem.

I published the source and release on codeplex:

http://montyhall.codeplex.com