In a random group of 23 people, what is the probability that at least two people share the same birthday?

Instruction: Assume 23 independent birthdays, each uniform over 365 possible days, and ignore leap day. Calculate the probability of at least one matching pair using the complement.

Context: This classic probability problem tests the candidate's understanding of the birthday paradox.

Updated

Official answer

Assume independent birthdays uniformly distributed over 365 days, ignoring leap day. To avoid any shared birthday, the successive people must use 365,364,…,343 available days. Therefore

P(at least one shared birthday) = 1 − product(i=0,…,22) [(365−i)/365] ≈ 0.507297, or 50.73%.

This concerns any matching pair among 23 people, not a match to one designated person's birthday. There are 253 pairs, which helps explain why the probability is substantial; their match events overlap, so simply adding 253 probabilities is not exact. Real birthdays can be seasonal or dependent, so the result belongs to the stated model.

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