The Monty Hall Problem: Switching IS Better!

Imagine yourself in a large circular room, totally empty, except for the presence of three doors. Just three unremarkable doors, completely identical in size, shape and colour. BUT WAIT. A note mysterious emerges from underneath the middle most door. Retrieving the note you read:

We three doors of hope and misery,

Behind one of us, your wildest desires will be,

But fail to choose the right one of three,

And only sorrow and loss will beget thee,

Your first thought upon reading the note is how unfortunate you are to have walked into some Hobbit-like riddling game. Mulling over the note once more, you reason that one door probably leads to something good and the other two doors probably lead to something bad. You surmise that the thing to do is pick one of the doors in hopes that, even if it conceals something bad, at least you'll be able to get out of the retched room.

You pick a door and just as you are reaching for the doorknob, you hear a loud bang and one of the other two doors bursts open, revealing a pile of sooty coal with a note tacked to the top of it. After scrambling up the pile of coal to get the second note you read:

Now two doors remain to choose from you see,

The exclusion of one sorrow I have granted for thee,

Stay or switch from your choice must ye,

Now choose for us which one it will be,

Do you stick with your original door choice or switch? You are staring blankly from one door to the next when suddenly an old statistics professor of yours materializes from behind the pile of coal. Sooty and slightly disoriented-looking, he walks towards you and exclaims “ There is a 2/3rd chance of winning if you switch doors but only a 1/3rd chance of winning if you stick with the original door!'

You think to yourself; Well that can't be right! Shouldn't the probability now be fifty-fifty? If only I had a . . .

'Pop', a computer with python 3 emerges into existence right in front of you. 'Ahah!' you exclaim, 'I can write a computer program that simulates the probability of winning given that I switch!'

At least, that is what I did when I encountered this problem, however less gloriously. This is informally known as the Monty Hall Problem, thus named for the host of an old game show which presents a similar scenario to its contestants.

There is a large body of theoretical evidence supporting the fact that switching doors IS better. But even after applying Conditional Probability, Bayes Theorem and the Law of Total Probability, I still found myself struggling to believe it. Being the CompStatNerd that I am, I decided to write out a program to simulate the problem. I will post my code here only because it is entirely of my own making and it is really quite simple.

I have set up the program to simulate the case where the individual always switches doors. The program tallies the number of total_wins and total_losses that result from the given number_of_games played. The doors are represented as an ordered list containing two goats and one car.

For each time that the game is played, one of the 'doors' (containing either 'goat', 'goat' or 'car') is chosen at random and removed from the list.

If the door chosen contained a car, then exactly two goats remain in the list. If the door chosen contained a goat, then exactly one car and one goat remain in the list. If the former is true, then one of the two goats remaining is chosen at random and is removed from the list. This list now contains only one goat. If the latter is true, then the only goat in the list is removed. This list now contains only one car.

A win is scored if the resulting list contains the car. This is because the contestant had initially chosen a goat and now, upon switching, has chosen the car. A loss is scored if the resulting list contains a goat. This is because the contestant had initially chosen the car and now, upon switching, has chosen the goat.

from random import randint

number_of_games = 100000

total_wins = 0

total_losses = 0

for i in range(number_of_games):

#Pick a door.

#replace a random value in list doors with 'choice'.

doors = ['goat', 'goat', 'car']

choice = randint(0, 2)

doors[choice] = 'choice'

 

#Monty shows a goat.

#if list contains car, remove the only instance of goat.

#if list contains 2 goats, choose one at random to be removed.

if 'car' in doors:

doors.remove('goat')

else:

doors.pop(choice)

i = randint(0, 1)

doors.pop(i)

 

#Switch doors.

#If car is still in the list doors then a win is scored, else a loss is scored

if 'car' in doors:

total_wins += 1

else:

total_losses += 1

print('Total wins: ' + str(total_wins))

print('Total losses: ' + str(total_losses))

Upon running the program with 100 000 games, the following output resulted:

Total wins: 66878

Total losses: 33122

This means that by switching doors the contestant won 66878 times out of 100 000 which roughly corresponds to winning 2/3rd of all games. WOW!

Even though every fiber of my being hates to admit it, it turns out that switching IS better!

It just goes to show that when dealing with probabilities, our brains (my brain at least) is ill-equipped to understand what is really going on. But we must strive to understand what is actually going on because it can help guide us towards making sound decisions (Dr. N Taback, personal communication, January 16, 2014). Through the use of tools like statistics, math and computation we can be confident that we have found the kernel of truth beneath our surface perceptions.