Games: goats and cars: the final episode

I had hoped not to have to return to the topic of goats and cars, but the amount of post I have received on the subject has made a final visit inevitable. Statisticians, mathematicians and accountants have written in to tell me that it's all perfectly obvious, and I should never have been taken in by the paradox, and they don't see what the problem is. But even more probabilists, actuaries and goat-lovers have written to say that we've all got the answer wrong anyway. So here, back by public demand, is definitely the last appearance of the Great Goats and Cars paradox. Let me take a deep breath and see if I can finally explain it.

To recap: A quizmaster offers a contestant the choice of three doors. One hides the star prize of a new car, the other two conceal only goats. Contestant selects door; quizmaster (who knows where the car is hidden) then opens one of the other two doors to reveal a goat. Then he offers the contestant the chance to change his mind. Should he do so?

Most people say it makes no difference. The choice is now reduced to two doors and there's a one-in-two chance the car is behind each. WRONG, WRONG, WRONG! The truth is that a change of mind doubles your chance of winning. And for all who still do not see why, here are two final efforts to explain it.

1: After the door has been opened, you have two options: to change your mind or stick with your original choice. If you stick with your original choice, you will win if and only if your original choice was correct. If you change your mind you will win if and only if your original choice was wrong. Your original guess will be right only on one of three occasions; it will be wrong on two out of three. So changing your mind will secure the prize on two out of three occasions, while sticking to your original choice will locate the car only one in three times.

If that doesn't totally convince you, try this:

2: Look at the original choice not as a simple selection of one door, but as dividing the doors into two groups: Group A, the one door you chose; and Group B, the other two doors. The quizmaster now offers you the choice between Group A and Group B. If you pick Group A, you win the car if it's behind that one door; if you pick Group B you win the car if it's behind either door in the group. Put that way, its perfectly obvious that you should pick Group B - you have twice the chance because it has two doors rather than one.

But that's exactly the same as the game we started with. Only by eliminating one of the doors in Group B, the quizmaster makes it look just like Group A - two identical-looking doors - but because of the preamble, they are far from probabilistically equivalent.

If anyone is still unconvinced, I can only suggest that they get someone else to play the game with them, hiding something at random under one of three cups. Try both strategies and you will soon see that changing your mind produces results twice as good as not doing so.

And to the many who wrote to chastise me for being bemused by a perfectly simple probabilistic calculation, let me plead that you have misunderstood my bemusement. What perplexes me is the psychological aspect of how easy it is to be taken in by this paradox. The answer, as most of the post- bag has confirmed, is definitely counter-intuitive, but I still don't completely understand why.

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