Monty's Dilemma -- Let's Make A Deal

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There is a TV game show in which the contestant is asked to choose one of three doors. Behind one of the doors is a whopping big prize, and behind the other two doors there is junk.

After the contestant chooses one of the doors, the game show host shows him what is behind one of the other two doors, always showing a "junker." Then the contestant is presented with the following dilemma:

Would you like to keep the door you chose, or switch to the other ( still veiled) door?

Monty's Strategies

Let us pose a mathematical (probabilistic) problem from this dilemma

Which of these three strategies is most likely to lead the contestant to the winning door?

1) Just stay put, and keep the original door you chose, after the junk door is disclosed.

2) Choose again by randomly selecting a door from the remaining two closed doors.

3) Choose again by switching from the door you chose to the other closed door.

How could we simulate the playing of Monty's game many times over?

 

The answer is (3) - You should always switch!
The first door has a 1/3 chance and the second door has a 2/3 chance.

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This problem first appeared in the Ask Marilyn column in Parade magazine. It has also been featured in the Mathematics Teacher and other math journals.   Many professional mathematicians wrote in to say that she was wrong.   Here is Marilyn vos Savant's reply to a reader who said that the odds would go up to 1/2:

Dear Readers,
    Good Heavens!   With so much learned opposition, I'll bet this one is going to keep math classes all over the country busy on Monday.

    My original answer is correct.   But first, let me explain why your answer is wrong.   The winning odds of 1/3 on the first choice can't go up to 1/2 just because the host opens a losing door.   To illustrate this, let's say we play a shell game.   You look away, and I put a pea under one of three shells.   Then I ask you to put your finger on a shell.   The odds that your choice contains a pea are 1/3, agreed?

Then I simply lift up a shell from the remaining other two.   As I can (and will) do this regardless of what you've chosen, we've learned nothing to allow us to revise the odds on the shell under your finger.

    The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities.   For the first three games, you choose #1 and "switch" each time; for the second three games, you choose #1 and "stay" each time; and the host always opens a loser.   Here are the results: