Asked in Probability
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?
September 17, 2007 4:46AM
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 ]
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%