Birthday Paradox

How many people are required so that the probability is greater than 50% that two of them share a birthday. Many people think that dozens, even hundreds would be required, but in any group of 23 or more people, the probability is greater than 50% that two share a birthday. This is known as the birthday paradox (the word "paradox" is used here not in the sense of a self-contradictory statement but in the sense of an unintuitive but true statement). This fact may come as a surprise to many people, but it's not hard to show mathematically.

First off, the probability that two people share a birthday is equal to 1 − (the probability that everyone has a different birthday).

• For a room containing one person, the probability that everyone has a different birthday is 1.
• For a room containing two people, the probability that everyone has a different birthday is 365/366, or approximately 99.7%. If no-one is to share a birthday, person #1 can have any birthday whatsoever, and person #2 can have a birthday on any of the 365 days that are not that birthday.
• For a room containing three people, the probability that everyone has a different birthday is (365/366) × (364/366), or approximately 99.2%. If no-one is to share a birthday, person #1 can have any birthday, person #2 can have a birthday on any of the 365 other days, and person #3 can have a birthday on any of the 364 remaining days.
• ...
• For a room containing 23 people, the probability that everyone has a different birthday is (365/366) × (364/366) × (363/366) × ... × (345/366) × (344/366), which equals approximately 49.4%. Therefore, the probability that two people share a birthday is 100% − 49.4% = 50.6%.

In a group of 23 there are 253 possible pairs of people. When you think about the problem like that, the above begins to make sense.

In his book Lady Luck, the mathematician Warren Weaver relates how this curious fact came up in conversation during a dinner party for a number of army officers during World War II. Most of the people with whom Weaver was dining found it hard to believe that the number was just 23; they were certain it would have to be in the hundreds. Someone pointed out that there were 22 people seated around the table, so he put the theory to the test.

In turn, each of the guests revealed his birth date, but no two turned out to be the same. Then the waitress said, "Excuse me, I am the 23rd person in the room, and my birthday is May 17, just like the general's over there."

Here is a related lateral thinking puzzle of interest.

View a list of mathematical paradoxes.