Subject: Probability of birthday matches
Date: Sat, 6 Mar 2000 07:50:55 -0600 (CST)
There is a counter-intuitive problem which I'm sure you are well aware of, namely the probability of having at least one pair of individuals in a group sharing the same birthday. I understand that the underlying idea of this problem is that you have to count the number of ways people in the group can be paired up. Specifically, in a group of n, there are n*(n+1)/2 pairs of individuals. This one source states that with a group of 23, the probability of a shared birthday in the group is slightly more than 50%, stating that there are n*(n+1)/2=23*24/2=276 pairs of birthdays. What I am confused about is that it is possible (albeit remotely possible) to have a group of 364 individuals and still not have a shared birthday, yet in this case n*(n+1)/2=364*365/2=66420. My question is, how can 276 pairs have a probablity of 50+% and yet with 66420 pairs we still have not reached 100% probablity? How do I calculate the probablity for a given group n?