Alex Y

The birthday problem comes into play here. That problem says that the 23th person in line has a (slightly) better than 50-50 chance of having a birthday the same as one of the 22 people before them in the line.

Larry, I think the problem you are thinking of is the point at which there is a 50-50 chance that there are two people in the room that have the same birthday (any two people, not just the last person and one other). The probability that the last person entering the room will have the same birthday as one of the other 22 people is 1-(364/365)^22=~6%.

But you are right that this problem is closely related to that one, and the answer is close to (but not equal to) 23.