## Sunday, February 12, 2012

### It's a boy!

Here's an interesting problem posed to me by MAJ: A woman has two children. Given that she has a boy born on a Tuesday, what is the probability that she has two boys?

1. The answer depends on how, and why, that information was "given" to you. But in the original formulation of the problem, a man stood up in front of an audience of math puzzlers, and said "My name is Gary Foshee. I have two children, and one is a boy who was born on a Tuesday." He claims the answer is 13/27, but he is wrong. It is 1/2.

His mistake was thinking the condition is "all families of two, that include a boy born on a Tuesday." If that were the case, of 14^2=196 possible combinations of a gender and a day, 27 include a boy born on Tuesday (14 where the first-born qualifies, 14 where the second-born qualifies, and subtract 1 for double counting the case where both do). And 13 of them have two boys (same logic, 7+7-1). That would make the answer 13/27.

But suppose, as examples, Mr. Foshee also had a boy born on a Thursday, or a girl born on a Tuesday. In these cases, in order to formulate his puzzle, he would have to choose one combination from two possibilities. Since nothing in his statement implies he was required to pick the Tuesday boy, we must assume it was equally likely he would have picked the other combination. That is, there are cases where the facts apply but he would say something else. To account for choice, the contribution of all combinations except "two Tuesday boys" must be divided by two, and that makes the answer (1+12/2)/(1+26/2)=1/2. This also is the answer when "Tuesday" is omitted, and you are merely told that a two-child family includes a boy.

In your question, the answer is 13/27 if you were "given" this information because you specifically asked "Is one of the two children a boy born on a Tuesday?" But it is 1/2 if you found out by any other means. The reason it is not 1/3, which some people claim is the answer when the day of birth is not mentioned, is because a two-boy family is almost twice as likely to have a Tuesday boy, as opposed to a one-boy family. This change is unintuitive, because it is unintuitive to assume such an odd question was asked.

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3. Hey Tony,

Came across your blog by search for russian! Did you stick with it since you started back in 2006? :)

4. Alas, I didn't. My son is still studying it and doing very well.