Solution
Sharing Your Birthday
Show the problem again
Birthdays are independent and uniform over 365 days (ignore leap years). Your birthday is fixed. What is the smallest number n of other people for which the probability that at least one of them shares your birthday exceeds 1/2?
Worked solution
The answer is 253. Each other person misses your birthday with probability 364/365, independently, so we need 1 − (364/365)^n > 1/2, or n > ln 2 / ln(365/364) ≈ 252.65. Hence n = 253. This is far more than 23 because only n comparisons involve you, whereas a group of 23 contains C(23,2) = 253 pairs, each of which could match. The coincidence is that the two answers are linked by that count: 253 comparisons either way.
Source: Classic companion to the birthday problem in wide circulation; no traceable original source. Statement written for AxiomIQ.