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The Monty Hall Paradox

File:Monty hall abc tv.JPG


An old paradox, named after a game show host, the emcee of "Let's Make a Deal" continuously from 1963 to 1976, and intermittently as revived thereafter.

Monty Hall presented contestants with three curtains, often labeled A,B, and C, and asked them to make a choice. They would get whatever gift was behind the curtain they chose.

In the story/paradox, Monty tells you that behind one of the curtains is a magnificent new car, the car of your dreams. Behind the other two curtains, not-so-desirable gifts.

You tell him: "I pick A, Monty."

Monty: "Not so fast! I will now reveal what is behind curtain C."

The curtain opens, and there is some worthless gag gift, say a pile of shaving cream.

Monty: "Now, you can stay with your choice of A, or you can switch to B."

What should you do?

The usual first response, and for many people a very powerful intuitive response, is that it doesn't matter what you do. You now have just two curtains left, so the odds of being right either way are 1/2, right? So you can stay with A or go with B: it's all the same.

Another consideration: psychologically some people would probably be likely to stay with curtain A after the revelation as to C. After all, the revelation is consistent with your initial hypothesis. It may even be dignified with the word "evidence." So its evidence that you were right ... why change?

Well ... both of those responses are wrong. As a matter of simple game theoretical logic, the best thing to do is to switch -- go with curtain B now.

Why? Because Monty knows which curtain the prize is behind. The selection of C as the curtain to open was not random --he must have known the shaving cream was there. [If he didn't, he'd be at risk of selecting the curtain with the car behind it himself, ruining the whole bit.]  Since the selection of C wasn't random, it doesn't give you any new information.

Nor are your odds of being right 1 in 2. They were 1 in 3 when the game started and (because of the non-random nature of Monty's choice) they are still 1 in 3. As I say, there has been no new information.

The only rational thing to do is to switch to curtain B. There are now 2 chances in 3 that the car is there.

Its weirdly counter-intuitive, but perfectly logical. I love it so.  
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