Today is my birthday – I figured there was no better way to celebrate than to talk through the birthday problem with the boys.
We’ve talked about it before – here’s what they remembered:
Now . . . how to we tackle this problem? Getting started proved to be a little difficult:
Now that we figured out how to approach the problem, we dove into the calculations. This step also proved to be challenging, but eventually they were able to see how the calculation worked.
Finally we wrote a little [ bug filled ] program on Mathematica and played around with the results.
It is always fun to share a famous problem with the boys.
Mike's Math Page 
Here’s an interesting fact that’s not sufficiently well known: if there are n days in a year, and the random variable N is the number of people who enter the room until the first birthday-match occurs (or, if you prefer, the number of times we roll a fair n-sided die until we roll a number we’ve seen before), then the expected value of N(N-1)/2 is exactly n. I’d love to know a purely combinatorial (preferably bijective) proof of this fact.