Solution:

It is easier to see this as what is the chance that no one shares a common birthday? We can call this probability A. Therefore, our solution is the compliment, 1-A: at least two people or more have a common birthday. What is probability A? If there are two people, the chance that they don’t have the same birthday is (364/365); there is a 1/365 chance that they do have the same birthday. This only leaves 363 days left for our third person, making the probability that three people don’t share a birthday as 1*(364/365)*(363/365). If we continue with the math we see that we need 23 people in order to have a probability of 49.3% that no one shares a common birthday. Since we are searching for 1-A, this means there is a 50.7% chance of a common birthday if there are 23 people in the room.