Coincident birthdays
Probability
Level
3
How many people have to be in the same room for there to be more than half a chance that two of them have coincident birthdays (day and month identical)?
The answer is 23.
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1 solution
Take a person at random. S/he has a birthday. For person 2 to not be coincident, their birthday has to fall on one of the 364 other days. If this happens, for person 3 to not be coincident, their birthday now has to fall on one of 363 not used days. And so on.
So for N people, the chance that there is not any coincident birthdays is ( ( 3 6 5 × 3 6 4 × 3 6 3 . . . × ( 3 6 6 − N ) ) / ( 3 6 5 N ) )
This becomes less than 1/2, thus making the chance that there IS a coincident birthday more than half a chance, at N=23.