there are 25 students in a class room. determine the probability that 2 of them have the same birthday?

Question

365 days

Answer 1

The easiest approach is to compute the probability that no two students have the same birthday, and then subtract this probability from 1.

By the laws of conditional probability, (the probability that n students share no birthdays) equals (the probability that (n-1) students share no birthdays, times the probability that student n doesn't have the same birthday as any of the first n-1 students). Since those n-1 students have n-1 different birthdays, there are 365 - n + 1 birthdays that work for student n, out of 365 possibilities.

Since this probability is clearly 1.0 when n = 1, we get the following formula for the probability that no two students have the same birthday in a class of S students:

p = Product (n = 1 to S) of (365-n+1)/365.

This is a royal pain to compute by hand for S as large as 25. However, firing up Python real quick gives me that p ~= .431.

Thus, the probability you ask for = 1 - p ~= .569

The Python session to get this answer looks like this:

>>> p = 1.0
>>> for i in range(25):
... p*= (365-i)/365.0
>>> p
0.43130029603053599
>>> 1 - p
0.56869970396946401

Answer 2

I remember this in a high school statistics class. Answer is 50%. Just don't ask me to do the math to explain why...

Answer 3

It is close to 50%

Answer 4

100%

Answer 5

.137 or 13.7%