# Birthday problem-probability

Tachyon314
Most people have heard about the birthday problem if 23 people are placed in a room what is the probability that 2 people share the same birthday. Well, we can make 253 pairs and divide it by 364.

253/364= 69%
Anyway, this isnt my question.

This is:

Consider the following, 15 people are placed in a room, all of whom are born in January. What is the probability that 2 people have the same birthday?

However, using the same logic as the previous problem we end up with 105 pairs which is to be divided be 31.

Which isnt true.

Any ideas?

## Answers and Replies

Gold Member
253/364 doesn't make sense to me.

The probability that at least two have same date of birth in a room of n people is one minus probability that all have different birth day, that is

p(n) = 1 - 365/365 * 364/365 * ... * (365-(n-1))/365,

which gives p(23) = 0,507

I'm sure you can modify that to fit the "birthday in january paradox". I get that in this case n >= 17 gives pjan(n) >= 0,5.

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