If no two share a birthday then the first person can choose a birthday
in 365 ways, the second person in 364 ways, the third in 363 ways, and
so on. If there are n people in the room, the probability that no two
share a birthday is
365 x 364 x 363 x ... x (365-n +1) P(365,n)
---------------------------------- = --------
365^n 365^n
So the probability that two at least share a birthday is:
P(365,n)
1 - --------
365^n
.........................................................P(365,n)
If this must be >= 0.5 then we require -------- <= 0.5
..........................................................365^n
(Here, P(365,n) is the number of permutations of n things that can be
made from 365 different things)
There is no easy way to solve the equation for n, but with a
calculator which gives permutations and combinations, trial and error
will quickly establish that n must be around 23.
P(365,23)
--------- = 0.4927
365^23
and 1 - 0.4927 = 0.5073
and so 23 people are sufficient to make the probability of a shared
birthday greater than 50%. this means that 33 people are more than enough.
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