Probability of two birthdays happening on the same day

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?

Re: Probability of two birthdays happening on the same day

As far as I can tell, the second method is simply finding how many ways to have two people have the same birthday. It doesn't seem to relate to probability in the way that you think.

2 People

AB

3 People

AB

BC

AC

4 People

AB BC

AC BD

AD CD

And so on. I'm no expert on probability however, so my explanation may be flawed.

Re: Probability of two birthdays happening on the same day

Therefore if each of those combinations have a probability of occurring of 1/365 and all are independent of each other then you simply add them all up?

Re: Probability of two birthdays happening on the same day

Not really. You would find all 190 combinations (20nCr2), and that would give you all of the different ways two people (and only two people) could share a birthday. This does not tie into how likely each one is to happen. For that, you would have to use the mathematical method.

Re: Probability of two birthdays happening on the same day

So in a simpler example if two people each have a bag of 3 balls, red, green and blue. The chances of Person one getting a green ball is 1/3 and the second is 1/3 so why isn't the chance of either being green 2/3?

3 balls, two chances to get a green ball, and it resets each time 2/3?!