I certainly won't be the only one to catch your latest error [November 2], but the perverse joy I get pointing it out offsets the small chance this letter has of being printed. I refer to your answer to Michael Grice's question about the game show conundrum--one prize, three doors, you pick Door #1, the host opens Door #3 to reveal no prize. Should you switch to the remaining door or stick with your original choice? You agreed with Grice that the odds of winning are equal for both--one in two. Wrong! The easiest route to the truth is to notice that resolving never to switch is equivalent to not having the option to switch, in which case, I'm sure you'll agree, the odds of winning remain one in three. Switching, therefore, has a two-thirds chance at the prize.
Your mistake was not realizing that opening Door #3 tells you more about Door #2 than about the door you originally picked. The reason for this is subtle. The host, in picking Door #3, does not choose from the full set of doors but rather from the subset of doors you did not pick. Each subset's probability of winning does not change but the probability for a particular door in the second subset does. If you don't get it find a friend who looks like Monte Hall and play 20 rounds. It will soon become obvious which strategy wins most often. --Robert E. Johanson, Chicago
Not so fast, amigo. I'll admit I wasn't paying much attention when I wrote that column and fell into a sucker's trap. But now that I've had a chance to study the matter, it's apparent there is a subtlety that eluded you as well.
First, though, I feel obliged to eat a bit of crow. The "common sense" answer, the one I gave, is that if you've got two doors and one prize, the chances of picking the right door are 50-50. Given certain key assumptions, which we'll discuss below, this is wrong.
Why? A different example will make it clear. Suppose our task is to pick the ace of spades from a deck of cards. We select one card. The chance we got the right one is 1 in 52. Now the dealer takes the remaining 51 cards and turns over 50, none of which is the ace of spades. One card remains. Should you pick it? Of course. Why? Because (1) the chances were 51 in 52 that the ace was in the dealer's stack, and (2) the dealer then systematically eliminated all (or most) of the wrong choices. The chances are overwhelming--51 out of 52, in fact--that the single remaining card is the ace of spades.
Which brings me to the subtlety I mentioned earlier. Your analysis of the game show question is correct, Bobo, only if we make several assumptions: (1) Monte Hall knows which door conceals the prize; (2) he only opens doors that do not conceal the prize; and (3) he always opens a door. Assumptions number one and two are reasonable. Number three is not.
Monte Hall is not stupid. He knows, empirically at least, that if he always opens one of the doors without a prize behind it, the odds greatly favor contestants who switch to the remaining door. He also knows the contestants (or at least the highly vocal studio audience) will tumble to this eventually. To make the game more interesting, therefore, a reasonable strategy for him would be to open a door only when the contestant has guessed right in the first place. In that case the contestant would be a fool to change his pick.
But that's absurd, you say. If Monte only opened a door when you'd chosen correctly in the first place, no one would ever switch. Exactly--so it's likely Monte adds one last twist. Most of the time he only opens a door when you've chosen correctly in the first place--but not always. In other words, he tries to bluff the contestants, then counterbluff them.
This strategy changes the odds dramatically. In fact, it can be shown that if, two times out of three, Monte opens a door when the contestant has guessed right the first time--a very rational approach on his part--over the long haul the odds of the prize being behind Door #1 versus Door #2 are 50-50.
Probability isn't the cut-and-dried science you might assume from high school math class. Instead it involves a lot of educated guesses about human behavior. I'll admit I jumped to an unwarranted conclusion on this one. Don't be too sure you haven't done the same.
Art accompanying story in printed newspaper (not available in this archive): illustration/Slug Signorino.