The Birthday Paradox

It's my birthday today (25 is SO last season), which made me start thinking about the age-old Birthday Paradox. The question is, If you are in a room with 30 people, what's the probability that 2 people share the same birthday? I'm sure a lot of reasonable people will assume the chances are pretty low, say 10%. In reality, there is a 70% likelihood that there will be two people in that group of forty that have the same birthday.

How is this mathematically possible? Well, you have to remember that you are looking for the probability that ANY two people have the same birthday, not that two people share a specific birthday. It starts by figuring out the probability that everyone in the room has a different birthday. The first person can have any 365 days of the year. The second person can have any birthday except the first person's birthday - 364 out of 365 days. The third person can have any birthday besides the first two - 363 out of 365 - and so on. When you multiply these ratios together - 365/365 x 364/365 x 363/365 ... you will get the probability of people not having the same birthday. Subtract that number from one and voila! the Birthday Paradox comes to light. When we take out those pesky things like leap year babies and seasonal birth rates, the chance that two people will have the same birthday nears 99% for a group of 60.

Computer nerds have known about this little phenomenon for a while and have used it to decrypt codes. I probably won't become a code breaker anytime this year, but maybe I'll be able to impress my friends... or make money off of bets with them.