You’ve all heard this classic statistics problem, based on an old game show:

A contestant is shown 3 doors. Only one of those three doors hides something of value to the contestant (perhaps a new car), while the other two contain nothing. The contestant chooses one door, but that door remains closed. The host then opens up a 2nd door, and this door is *always* a losing door. At this point, the contestant may choose to now open the originally-chosen door, or switch to and open the last remaining door.

So why is this interesting? It turns out that the way to maximize your chances of winning is to *always switch*, and this maximized chance is 67%. It also turns out that this is totally non-intuitive, and that most people think that, if the contestant always switches, the chances of winning are at best 50%. If you haven’t heard the solution to this problem before, you should think through it and see what you expect the chances of winning are under the two conditions: After the contestant chooses a door, and is subsequently shown that one of the other two is a losing door, [1] the contestant *always* switches to the remaining door, or [2] the contestant *never* switches. After the jump, I’ll explain this intuitively and then show a Python script to simulate this problem.

Whether the contestant is *always *or *never *switching, the outcomes only rely on the initial choice (see Figs.1&2 below). The easy case is when the contestant does not switch. Since only 1 of the 3 doors is a winning door, the contestant will only choose that door 1/3 times, meaning that s/he will only win 1/3 (33%) of the time.

When the contestant always switches after the host’s reveal of a losing door, the outcome is still decided by the initial choice. Say the contestant chooses the only winning door (Fig.1). This happens 1/3 of the time. In that case, a switch will *always* be to a losing door. So, we already know that the contestant loses at least 1/3 of the time.

Now what happens when the contestant chooses a losing door (Fig.2)? This happens 2 out of 3 times. In that case, what remains will be the single winning door and the other losing door. However, the host always opens a losing door. Therefore, the host will show the contestant the only other losing door, and then only the winning door remains. If the contestant switches, s/he can only switch to the winning door.

So, when the contestant always switches, s/he will win every time the initial choice was wrong (2/3=67% of the time), and lose every time the initial door was correct.

Hopefully that makes intuitive sense. Now here’s the fun part. Say you wanted to expand this problem and say “given **N** doors, ** T** of which are winning doors, how does the probability of winning change when choosing to always or never switch?” We can model any case described in this way with a Python class:

### 2011 GNU GPLv3 by Adam Coster (adamcoster.com) ### ### Feel free to copy, change, and redistribute but, ### ### as always, give credit where credit is due! ### import random class Monty: """ A class representing an in instance of the Monty Hall problem.""" def __init__( self, true_doors, total_doors ): """ Initialize a Monty object, given the winning and total doors. Inputs: true_doors - an integer representing the number of "winning" doors total_doors - an integer representing the total number of doors Obviously, total_doors must be greater than true_doors """ self.true_doors = true_doors self.total_doors = total_doors # Set up a dictionary of i:False pairs, with each i representing one # of the total doors. Doors can then be chosen by an integer key self.doors = { i:False for i in range( total_doors )} self.set_true_doors() def set_true_doors( self ): """ Randomly choose self.true_doors keys in self.doors, setting value to True.""" for win in range( self.true_doors ): potential_winner = self.random_door() # The randomly-chosen door may already be set to true, in the # case where self.true_doors is > 1. There must be EXACTLY # self.true_doors values set to True: while self.doors[ potential_winner ]: potential_winner = self.random_door() self.doors[ potential_winner ] = True def choose_random_door( self ): """ Choose a random door, and set the value of self.win to its value.""" self.choice = self.random_door() self.win = self.doors[ self.choice ] def switch_choice( self ): """ Make one False door off-limits, and switch the chosen door to another.""" revealed = self.reveal_false_door() new_choice = self.random_door() while new_choice == self.choice or new_choice == revealed: new_choice = self.random_door() self.choice = new_choice self.win = self.doors[ self.choice ] def reveal_false_door( self ): """ Randomly choose a self.door key whose value is False.""" false_door = self.random_door() while self.doors[ false_door ] or self.choice == false_door: false_door = self.random_door() return false_door def random_door( self ): """ Return a random value between 0 and (self.total_doors-1).""" return random.randint( 0, self.total_doors-1 )

I tried to make the code describe itself, so I’ll leave you to figure out how it works. In the unlikely case that a few people ask for those details, I’ll then describe what this class does in more detail. Finally, here’s an implementation of the Monty class that will let you choose the number of doors, the subset that are winning doors, and the number of simulations to run. It will print the percentage of simulations that resulted in a win. If you aren’t familiar with Python and don’t know how to make this go, see my “getting started” post. After you have Python running, you can just make a text file, paste in the above and below code into that text file, and save it as “monty_hall.py” or similar. Then double-click (or open in IDLE) and you’re on your way!

def main(): total = int(input( 'Total number of doors: ')) trues = int(input( 'Number of winning doors: ')) trials= int(input( 'Number of trials: ')) if input( 'Switch after reveal?: ').upper()[0]=='Y': switch = True else: switch = False wins = 0 for n in range( trials ): monty_hall = Monty( trues, total ) monty_hall.choose_random_door( ) if switch: monty_hall.switch_choice( ) if monty_hall.win: wins += 1 print( 'Percent win:', wins/trials ); input() if __name__ == '__main__': main()

I LIKE THIS.

You know what I love about your website, Adam? It’s both interesting and actually useful.

What is the name of the module you use to display your sample code?

In wordpress, you just need tags:

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Man, I can’t figure out how to write that without it becoming syntax highlighting…