Calendar Boy

Gary Foshee presented this puzzle at the 2010 Gathering for Gardner:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

The first thing you think is ‘What has Tuesday got to do with it?’ Well, it has everything to do with it.

He proposed the answer 13/27, with this reasoning:

There are 14 equally likely possibilities for a single birth — (boy, Tuesday), (girl, Sunday), and so on.

If all we knew were that Foshee had two children, then it would seem that there are 142 = 196 equally likely possibilities as to their births.

But we know that at least one of his children is a (boy, Tuesday), and only 27 of the 196 outcomes meet this criterion. (There are 14 cases in which the (boy, Tuesday) is the firstborn child and 14 in which he’s born second, and we must remove the single case in which he’s counted twice.)

Of those 27 possibilities, 13 include two boys — 7 with (boy, Tuesday) as the first child and 7 with (boy, Tuesday) as the second child, and we subtract the one in which he’s counted twice. That, Foshee says, gives the answer 13/27.

This generated a lot of discussion when it appeared — unfortunately because the meaning of Foshee’s question is open to interpretation. See the end of this New Scientist article and the comments on Columbia statistician Andrew Gelman’s blog.