What is the smallest number of people in a room where the probability of two of them having the same birthday is at least 50 percent?

23. The probability that at least two people in a room share a birthday can be expressed more simply, mathematically, as 1 minus the probability that nobody in the room shares a birthday.

Imagine a fairly simple example of a room with only three people. The probability that any two share a birthday is :

1 - [ 365/365 x 364/365 x 363/365]

i.e. 1-P(none of them share a birthday)

=1 - [ (365x364x363) / 3653 ]

=0.8%

Similarly,

P(any two share a birthday in a room of 4 people)

= 1 - [ 365x364x363x362 / 3654 ] = 1.6%

If you keep following that logic eventually you get

P(any two share a birthday in a room of 23 people)

=1 - [(365x364x...x344x343) / 36523 ] = 51%