Sharing the Same Birthday Here is a famous example that you can use to test your intuition about surprising events. How many people would need to be gathered together to be at least 50% sure that two of them share the same birthday (month and day, not necessarily year)? Most people provide answers that are much higher than the correct answer, which is that only 23 people are needed.

There are several reasons why people have trouble with this problem. If your answer was somewhere close to 183, or half the number of birthdays, then you may have confused the question with another one, such as the probability that someone in the group has your birthday or that two people have a specific date as their birthday.

It is not difficult to see how to calculate the appropriate probability using our probability rules. Note that the only way to avoid two people having the same birthday is if all 23 people have different birthdays. To find that probability, we simply use the rule that applies to the word and (Rule 3), thus multiplying probabilities. The probability that the first three people have different birthdays is the probability that the second person does not share a birthday with the first, which is 364/365 (ignoring February 29), and the third person does not share a birthday with either of the first two, which is 363/365. (Two dates were already taken.)

Continuing this line of reasoning, the probability that none of the 23 people share a birthday is

The probability that at least two people share a birthday is the probability of the complement, or + 2 .493 = .507.

If you find it difficult to imagine that this could be correct, picture it this way. Imagine each of the 23 people shaking hands with the other 22 people and asking them about their birthday. There would be 253 handshakes and birthday conversations. Surely, there is a relatively high probability that at least one of those pairs would discover a common birthday.

By the way, the probability of a shared birthday in a group of ten people is already better than one in nine, at .117. (There would be 45 handshakes.) With only 50 people, it is almost certain, with a probability of .97. (There would be 1225 handshakes.)