In a group of 23 people, the probability that 2 of them share a birthday is 50%. This fact may seem strange at first, but can be proven by fairly simple mathematics.

#### Combinations

In a group of 23 people, there are 253 unique pairings which can be made. This can be calculated by finding ^{23}C_{2}.

#### Probability

Assuming that each day in the year has an equal probability of being somebody’s birthday, the probability two people share a birthday will be 1/365.

This means the probability that they do not share a birthday will be 364/365. For nobody in the group of 23 to have a shared birthday, all 253 pairings will have to not share a birthday. As this is an AND probability, all these probabilities must be multiplied together.

Therefore, the probability that nobody in the group shares a birthday is:

P = (364/365)^{253} = 0.4995

Overall, the probability that two or more people have a shared birthday will be just over 50% (0.5005)

#### What if there are 75 people?

Applying the same method, it can be shown that there is a 99.95% chance that 2 people share a birthday out of a group of 75.

^{75}C_{2} is equal to 2775.

1-(364/365)^{2775} = 0.9995