## Even Odds

January 30, 2013

The laws of probability, so true in general, so fallacious in particular.
–Edward Gibbon

A puzzle has recently been brought to my attention. It goes like this: “I have two children. One of them is a boy born on a Tuesday. What is the probability that I have two boys?” This puzzle was posed to an audience at the Gathering for Gardner meeting in Atlanta, Georgia, by Gary Foshee. He went on to say, “The first thing you think is ‘What has Tuesday got to do with it?’ Well, it has everything to do with it.”

I disagree. The day the boy was born is utterly irrelevant. Perhaps it would be best to look at the “accepted” solution from New Scientist before I tell you why it’s hogwash. Its main point is:

The main bone of contention was how to properly interpret the question. The way Foshee meant it is, of all the families with one boy and exactly one other child, what proportion of those families have two boys?
To answer the question you need to first look at all the equally likely combinations of two children it is possible to have: BG, GB, BB or GG. The question states that one child is a boy. So we can eliminate the GG, leaving us with just three options: BG, GB and BB. One out of these three scenarios is BB, so the probability of the two boys is 1/3.