Problem of the week

This one has taken a bit of thinking. It's completely counter-intuitive, until you think about it the right way:

I have two children. At least one of them is a boy born on a Tuesday. What is the probability that I have two boys?

It's been doing the rounds on internet forums, and even made it into the Sunday Times. You have to ask yourself: what does each sentence actually tell me? Does it add any extra information? The first step is recognising how much information declaring one son contains. Quite a bit in this case. If I simply told you I had 2 children, I have a 1 in 4 chance of having two boys. Knowing that I have one son, we can now reassess that - giving me a 1 in 3 chance of having two boys. (Not 1 in 2 as seems obvious - think about the options, I have a 1 in 4 chance of having, in turn: A son and a daughter, a daughter and a son, two sons, and two daughters. Eliminating one of these equally likely options (daughter-daughter) gives me a 1 in 3 chance of each of the others. See: Monty Hall Problem)

The next step is to ask: What does knowing that one son was born on a Tuesday tell me? Not a lot, it seems. But this is deceptive, it actually tells you more than you think. Of course your son had to be born on a day of the week, and any day is equally likely. But knowing this fact means that we have one clear piece of information: If I have two sons, then clearly I have more chance of having a son born on a Tuesday, but how much more? I'll let you Google the answer for now, but it's surprising.