Triple birthday

One class above India's, the K class, has 16 pupils. Three of them have the same birthday. Now how probable is that?

The quick answer is that it is a lot more probable than you would think, but highly unlikely still.

Most people intuit it from their own perspective that probability of someone having the same birthday as me are very small. In that, they are right: You need at least 253 people to have a 50% chance of finding an other person who shared yours.

However, when you compare everyone in a group against everyone else, the odds change quickly: For a 50% chance of getting two same birthdays, you do not need more than 23 people. You only need 57 people for 99% probability making it almost certain of finding a paired birthday in any group of that size. For a small group of 16 pupils the chance is still 27%.

Chances of having a triple birthday in a group of 16 are a lot smaller: 0.37%. In other words, highly improbable. To reach a 50% probability, you'd need at least 90 people which is not that many.

Let's step back and think what are the assumptions that the estimate is based on. These kind of calculations or simulations all assume that birthdays are uniformly distributed around the year. We know for a fact that this is not the case here. There are several children who entered school a bit younger than average and moved together to upper class after redoing the reception class. Their birthdays are clustered to Autumn. Then there is the effect of the weekday. Most of the children are from abroad where giving birth during weekend is avoided when possible. It just too expensive to keep a full staff at hospitals outside normal working hours.

All these, and many more, reasons skew the distribution of birthdays quite far from uniform. Just how skewed it is, is very difficult to estimate. But when you start from 0.37% probability, the whole even will remain quite improbable no matter what. But as we now, improbable does not mean impossible!

Further reading:

http://en.wikipedia.org/wiki/Birthday_problem - but my analytical skills failed me, so I had to turn to simulation: http://www.mste.uiuc.edu/reese/birthday/. I modified this simulation programme to estimate the probability of triple birthdays.