If you have two people in a room, the probability that they share a birthday is really, really small. On the other hand, once there are 400 people in a room it's absolutely certain that some two of them will share a birthday. So as you add people to the room, the probability of a shared birthday, any shared birthday, goes from nearly zero right up to 1.

  • What is the smallest group size that would give a better that even chance of two people sharing the same birthday?

For a group size of n, the probability that all people have a different Birthday = $\frac{364}{365}\times\frac{363}{365}\times\frac{362}{365}...\frac{366-n}{365}$ = p

The probability that at least one pair share a birthday = 1 - p

If there are 23 or more (randomly chosen) people in a group then there is a more than even chance that two will share a birthday

Number
of People 		Chance of
no pairs 		Chance of
at least
one pair
1 1 0
2 0.997 0.003
3 0.992 0.008
4 0.984 0.016
. ... ...
22 0.524 0.476
23 0.493 0.507

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