We have seen earlier that the probability of two people sharing a birthday in a group of 40 is about 90%. And that is surprising to many of you. Part of the reason is that you start imagining someone sharing their birthday with yours. Well, that is quite another problem.
Probability of some sharing My birthday
The probability of someone sharing my (my, in inverted comma) is different. This problem is not as random as the previous one, as at least one person (i.e., me) is fixed! Let’s start with the inverse problem.
What is the probability that no one else has your birthday in a group of n people? In a group of 2, the chance is (364/365), and for 3, it is (364/365)*(364/365), and so on. In general, for a group of n people, the chance is (364/365)n-1. Since the probability you share a birthday with one and the probability that you do not share are mutually exclusive events (both can not happen at the same time), you subtract one from 1 to get the other, or the answer is 1 – (364/365)n-1.
For n = 10, the probability is 2.4%; for n = 100, it becomes 24%. To get a 50% chance that someone is sharing your birthday, you need 253 people in the room!
