Birthday Paradox

Why does 23 people give about a 50% chance?

We assume there are 365 possible birthdays (ignoring leap years), and each birthday is equally likely and independent.

Instead of directly calculating the chance of a match, we calculate the opposite event: that all people have different birthdays, then subtract from 1.

Step 1: First person

The first person can have any birthday: probability = 365/365 = 1

Step 2: Second person

To avoid a match, they must NOT share the first person's birthday: probability = 364/365

Step 3: Third person

They must avoid the two previous birthdays: probability = 363/365

General pattern

For the n-th person, the probability they do not match any previous birthday is:
(365 − (n − 1)) / 365

So the probability that all n people have unique birthdays is:
(365/365) × (364/365) × (363/365) × ... × (365 − n + 1)/365

Why it becomes ~50% at n = 23

As n increases, you multiply many fractions slightly less than 1. This causes the probability of all unique birthdays to shrink quickly.

At n = 23:
probability(all unique) ≈ 0.4927
so probability(at least one match) = 1 − 0.4927 ≈ 0.5073 (≈ 50%)

The key reason this grows so fast is that the number of possible pairs is:
n(n − 1)/2
For 23 people, that is 253 pairs, and each pair is a chance for a match.