Imagine you are standing in line with 35 other people, and to pass the time, the guy in front of you proposes a wager. He’s willing to bet $50 that no two people in line share a birthday. Would you take the bet?
If you’re like most people, you wouldn’t. With 36 people and 365 possible birthdays, it seems like there wold only be about a one-in-ten chance of a match, leaving you a 90% chance of losing $50. In fact, you should take the bet, since you would have better than an 80% chacne of winning $50. This is called the Birthday Paradox (though it’s not really a paradox, just a surprise), and it illustrates some of the complexities involved in groups.
Most People get the odds of a birthday match wrong for 2 reasons. First, in situations involving many people, they think about themselves rather than the group. If the guy in line had asked, “What are the odds that someone in this line shares your birthday?” that would indeed have been about a one-in-ten chance, a distictly bad bet. But in a group, other people’s relationship to you isn’t all that matters; instead of counting people, you need to count links between people.
If you’re comparing your birthday with one other person’s, then there only one comparison, which is to say only one chance in 365 of a match. If you’re comparing birthdays in a group with two other people – you, Alice, and Bob, say – you might think you’d have two chance in 365, but you’d be wrong. Thre are three comparisons: your birthday with Alice’s, yours with Bob’s, and Alice’s and Bob’s with each other. With four people, there are six such comparisons, half of which don’t involve you at all; with five, there are then and so on. By the timeyou are at 36 people, there are more than 600 pairs of birthdays. Everyone understands that the chance of any two people in a group sharing a birday is low; what they miss is that a count of “any two people” rises much faster than the number of people themselves. This is the engine of the Birthday Paradox.