I was having a discussion with TonyC this morning about birthdays and he pointed out that his office had a statistical anomaly wherein nine people in his office of 32 had birthdays in August. I started to try and figure out what the odds of that happening were, but I got stuck.

How would you go about figuring those odds?

The first thing I looked at was the odds of one person having a birthday in August. To simplify, let's ignore the length of months and any non-even birth distribution, and assume that it's 1 in 12. And then it seems that if you have two people, the odds raise to 2 in 12, and so on. But this can't be the case, as if you have 12 people, that would imply that the odds were 12 in 12, and it's perfectly reasonable for a group of 12 people to have no one born in August. I'm guessing that this mathematical analysis is just flat-out wrong.

So maybe we have to look at the odds of not being born in August. That would be 11 in 12 for an individual. And then square that for two people, which yields something very close to 10 in 12, and then the next power and the next power until you get to the 12th power, which comes out to about 35.2%, which would mean that in a group of 12, there's about a 65% chance that someone was born in August. That sounds more reasonable. If we take that to 32 people, that's about a 94% chance that (at least) one person was born in August, which, again, sounds reasonable.

But now how do we calculate the odds of multiple people being born in August? If the above algorithm, O(n)=(1-(11/12)^n)), is correct, does it make sense that the odds of two people in a group of 32 would be the odds of one person in that group times the odds of one person in the remaining group? That sounds reasonable, but using that formula, aren't we just finding the odds of someone being born in a random particular month, and not specifically August? How do we tie the results of the first set of odds to the next set?

It's at this point that my brain shuts down.

Help?