Okay, I majored in English, so probably do not qualify for the the geekiness competitions that sometimes break out over at the science blogs. But my wife will tell you that I work math problems for recreation, more or less the way others might do sodoku or crosswords. Lately I've been using for a source Mathproblems.info. Problem 16 posits a box with two coins, one "fair" and the other two-headed, and asks what is the probability that, if a coin is drawn that displays a head, the reverse side of that coin is a head as well.

I can easily satisfy myself that the answer is 2/3 by observing that a single trial can have four equally likely results: one side of the two-headed coin, the other side of the two-headed coin, a head on the "fair" coin, and a tail on the "fair" coin. So you get a head displayed three times, and, of those three, you are holding the two-headed coin twice. Hence, 2/3.

The problem is probably less interesting than the host's hilarious account of the mail generated by this problem and his claim that the answer is 2/3. It seems like very many people, observing that a result of heads does not preclude the possibility that the "fair" coin has been drawn, insist that the answer must be 1/2. Of course, 2/3 acknowledges that the "fair" coin may have been drawn, and the host patiently illustrates with an example concerning mislabeled soup cans. Suppose there are 2 million cans of tomato soup, half correctly labeled but the other half labeled "chicken noodle," and that one store receives 999,999 correctly labeled cans and one bad one, while another store receives the other 1 million, all but one incorrectly labeled. If you went to one of the stores, opened a can, and found it to be correctly labeled, would you ignore that piece of information when trying to decide which store you most likely were in? So why ignore an analogous piece of information when trying to decide what coin you are most likely holding?

Nevertheless, one of the host's correspondents, a high-school biology teacher, could not persuade his principal, still less many of his students, that the correct answer to the coin problem is 2/3. He recruited the host to remonstrate with the principal, which the host did, finally persuading the principal to relent, though in grudging fashion festooned with weasel-words. Check it out.

In my next post I plan on describing what Steven Pinker, author of How the Mind Works, says about episodes such as this one. It turns out that the mind scientist, though up-to-speed on the math, has considerable sympathy for the principal.

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