What is the probability that 2 people out of 20 have the same birthday?

Question

Please provide an easy probability formula to determine the probability that 2 people out of 20 have the same birthday.

Answer 1

Normally, we ask what is the probability that at least two have the same birthday. The assumption is made that all birthdays are equally likely and we have 365 days.

Then, it is 1 - P(all have different birthdays) = 1 - P(365,20)/365^20, as P(365,20) gives the number of possible orders when you can select 20 items from 365 without replacement. P stands for permutation.

In Excel, we get

1 - Permut(365,20)/365^20 = 0.4114

If we want exactly two, that is a slightly different question.

The number of single matches among the 20 people would be C(20,2) to choose the matching pair, then Permut(365,19) to select first the day for the pair, then the days for the 18 remaining.

C(20,2)*Permut(365,19)/365^20 = 0.3232

Answer 2

Probability that in a room of n people, two people have the same birthday:

365!

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365*(365-n)!