Happy birthday to you.......

Here's a hoary old chestnut, but if you haven't heard it before the answer might surprise you. If we (a) ignore February 29th, and (b) assume that a person is as likely to be born on any date as on any other date, then how big a group of people picked at random do we need for there to be a better than even money chance that at least two of them share the same birthday? Most people reason something like this - ignoring February 29th, there are 365 possible birthdates, so the answer must be somewhere around 182 or 3. In fact the answer is a surprisingly small 23!! The point is that what was asked for is the probability that two or more people share the same birthday - whatever that date is. If the question had been how big a group do you need before there's a better than 50-50 chance that two or more of them were born on March 2nd for example, or any other given date, then the answer would in fact be an equally surprisingly large 253 - the reason this is so much greater than 182.5 is because, as demonstrated above, it is virtually certain that many of them will share the same birthday!