Don’t get the goat

There used to be an American game show, hosted by Monty Hall, in which the guest had to choose one of three doors. Behind one was an expensive prize — a sports car, say. Behind the other two were booby prizes — goats.

After the contestant had chosen, Hall would open one of the other doors to reveal a goat. (With two doors to choose from, he could always do this — he knew where the car was.) He would then offer the contestant the chance to change their mind and choose the other unopened door.

Hardly, any one took this opportunity — perhaps, with good reason, as I will eventually explain. But, for the moment, let’s take the problem at face value, and assume that the car has equal probability (one in three) of being behind any given door. We will assume also that every one knows ahead of time that Hall always offers the contestant a change to change their mind, after revealing a goat. Should they change?

The argument against goes like this: the two remaining doors are equally likely to conceal a car or a goat. Since the odds are fifty-fifty, there’s no reason to change.

Or, is there?

(thanks to Prof Ian Stewart for such a nice little puzzle in probability theory).

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