Fun Fact - Birthday Paradox

A high school math class has 27 students. The teacher asks each student in turn for his or her birthday (just the day not the year), and writes them on the board. Soon a duplicate appears. Yesw, two students out of 27 are likely to have the same birthday, even though there are 365 days to choose from. How is this possible?

This is called the birthday paradox, though it is not as mysterious as the word paradox might suggest. 23 people gives you an even chance of finding a duplicate birthday. A few more, and the odds are definitely in your favor.

The math is easy if you turn the problem around. What are the odds that every birthday is unique - that there are no duplicates? The first one is free. The second person has a new birthday with probability 364/365. The third person has a new birthday with probability 363/365, then 362/365, and so on. Multiply these fractions together and the odds steadily decrease. By the time you get to 343/365, the ratio is 49.27%. With 23 people you have a 50 50 shot at finding a duplicate birthday. Move up to a class of 27 and there is a duplicate birthday with probability 62.7%.

If you have a roulet wheel with n different numbers on it, so that each spin of the wheel selects a number at random, you need to spin the wheel approximately sqrt(n) times to get a duplicate.

Further Reading

Combinatorics and the birthday paradox