The Two Children Problem Reconsidered

People have threatened to punch me for bringing up the Monty Hall Problem at parties, despite the fact that the people I brought it up to were quants (who presumably enjoy such things), and despite the fact that I don't otherwise make people do party games. I feed them, I talk with them, but I don't make them play Pictionary, 'cause that's just mean.

In the mean time, here's another troublesome problem, the Two Children Problem, posed by Gary Foshee

Gary Foshee, a collector and designer of puzzles from Issaquah near Seattle walked to the lectern to present his talk. It consisted of the following three sentences: "I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

Now, the answer we arrive at might be any of the following: 1/2, 1/3, or 13/27, depending on the method used to arrive at the answer. But what actually determines the correct answer is why you told us about the boy being born on Tuesday.

So why does intuition seem to lead us so astray? Both the intuitive and the mathematically informed guesses are wrong. Are human brains just badly wired for computing probabilities?

Not so fast, says probabilist Yuval Peres of Microsoft Research. That naïve answer of 1/2? In real life, he says, that will usually be the most reasonable one.

Everything depends, he points out, on why I decided to tell you about the Tuesday-birthday-boy. If I specifically selected him because he was a boy born on Tuesday (and if I would have kept quiet had neither of my children qualified), then the 13/27 probability is correct. But if I randomly chose one of my two children to describe and then reported the child’s sex and birthday, and he just happened to be a boy born on Tuesday, then intuition prevails: The probability that the other child will be a boy will indeed be 1/2. The child’s sex and birthday are just information offered after the selection is made, which doesn’t affect the probability in the slightest.

...Peres says that we shouldn’t despair about our probabilistic intuition, as long as we apply it to familiar situations. The difficulty of these problems is rooted in their artificiality: In real life, we almost always know why the information was selected, whereas these problems have been devised to eliminate that knowledge. “The intuition develops,” he points out, “to handle situations that actually occur.”