How many people before someone shares your birthday?

Question

With 23 random people in a room, the chance that some two of them share a birthday is just over one half. That is the birthday paradox.

How many other people must be in the room for the chance that at least one of them shares your birthday to exceed one half? Ignore leap years, and assume birthdays are uniform.

Then explain, in one sentence, why this number is so much larger than 23.

Accepted Answer

253 people. The complement. Each other person fails to share your birthday with probability 364/365, independently. With n others, the probability that none of them matches you is (364/365) to the power n. You want this below 1/2. Solve. Taking logarithms, n × ln(364/365) is less than ln(1/2), so n is greater than ln 2 / ln(365/364), which is 0.6931 / 0.002743, about 252.7. So n = 253: with 252 people the no-match probability is 0.5008, and with 253 it is 0.4995. Why so many more than 23. In the classic version there are many pairs that could match: 23 people form 253 pairs. (It is not a coin…

How many people before someone shares your birthday?

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