The Birthday Paradox

The Birthday Paradox is not a real contradiction. It is a contradiction between probability and intuition. Imagine a room with 23 people. What is the probability that at least two of them share the same birthday? Most people expect the probability to be small because there are 365 possible birthdays.

The surprising answer is a little over 50 percent, if we ignore leap years and assume birthdays are spread evenly through the year. The reason becomes clearer when we stop comparing one person with the calendar and start counting pairs of people. With 23 people, there are 253 different pairs. That is a lot of opportunities for two birthdays to match. It is easier to calculate the opposite probability first: the probability that everybody has a different birthday.

The first person can have any birthday. The second must avoid that one day. The third must avoid two days, and so on.

Multiplying these probabilities gives the chance that all birthdays are different. Then we subtract that result from 1. The number rises surprisingly quickly. With around 50 people, the chance of a shared birthday is already very high. I like this example because nothing mysterious is happening.

Our intuition simply counts the problem in the wrong way. We think, “There are 365 days, so 23 people is not many.” Probability thinks, “How many pairs are being compared?” This is a useful lesson in statistics and everyday reasoning. A situation can feel unlikely because we are asking the wrong version of the question. Sometimes changing the question changes our intuition completely.