The car and the goats

1. THE PROBLEM. A TV host shows you three numbered doors, one hiding a car (all three equally likely) and the other two hiding goats. You get to pick a door, winning whatever is behind it. You choose door #1, say. The host, who knows where the car is, then opens one of the other two doors to reveal a goat, and invites you to switch your choice if you so wish. Assume he opens door #3. Should you switch to #2? I'll call this Game I. It appeared in the Ask Marilyn column in Parade (a Sunday supplement) [4(a)]. Marilyn asserted that you should switch, arguing that the probability of winning, originally 1/3, had now gone up to 2/3. ("Marilyn" is standard terminology.) This led to an uproar featuring "thousands" of letters, nine-tenths of them insisting that with door # 3 now eliminated, #1 and #2 were equally likely; even the responses from college faculty voted her down two to one [4(b, c), 3]. There is no denying that the problem is tricky (even though, technically speaking, it involves only undergraduate mathematics). The purpose of this article is to unravel it all.