It was two peoples birthdays yesterday in my team at work of 20 people. We started to have a debate on what the probability of that happening. Mathematically, you'd work out 1 minus the probability of it not happening. ie;

1 - (364/365) * (363/365) ... (346/365) = 0.41

However logically, the odds of 2 people (A and B) having the same birthday is 1/365. If you introduce a third person (C) then A being as same B =1/365, A same C = 1/365 and B same C = 1/365 or 3/365. If you introduce a fourth person you have 6 combinations therefore the odds are 6/365. The combinations for 20 people are 190 but 190/365 is 0.52?

Obviously this method is flawed as a team of 28 people have 378 combinations which would give a probability greater than 1!

Why is the second method wrong, what am I missing?