How to Count: Birthdays and Lotteries

Abstract

There is a very famous problem about birthdays showing how the answers to certain problems can defy our intuition. The problem can be phrased this way: suppose you are at a party, in a hall filled with people. How many people do you think have to be present before the probability that at least two people have the same birthday is about 1/2? Having the same birthday here means the month and day must match; the year is irrelevant. Suppose, for example, there are 30 people at the party and someone comes over to you claiming at least two people there have the same birthday. You know he is a stranger to the group—he has no inside information. He wants to bet $10 that he is right. Is this a good bet to make? If you take this bet you will win only if everyone in the hall was born on a different day of the year. Since there are 365 days in the year and only 30 people present, it might seem quite likely that there are no repeats of birthdays in the place and that the $10 bet would be quite favorable to you.