Probability screen, often asked with the follow-up 'estimate it in your head'.
Shared birthdays
The question
How many people do you need in a room before the probability that two share a birthday exceeds one half?
Have a go before you open the solution. If you are stuck after a few minutes, the first paragraph below is usually enough of a nudge.
Show the worked solution
The answer
23.
Compute the complement. With n people, the probability all birthdays differ is the product over k from 0 to n-1 of (365 - k)/365.
Approximate each term as exp(-k/365), so the product is roughly exp(-n(n-1)/730).
Set that equal to 1/2: n(n-1) is about 730 times ln 2, which is about 506, so n is about 23.
The reason the answer feels small is that n people generate n(n-1)/2 pairs, and it is pairs that collide, not people.
What it is testing
Complement counting and a fast exponential approximation without a calculator.
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Preparing properly
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