9 people in a room what is the probability two of them share a birthday

Question

how to set it up

Answer 1

Ahhh, a classic.

 

Well, the way the problem is worded, I'll assume that we are looking for the probability that at least two of them share a birthday.  So, two people could have the same birthday, or three of them could have the same birthday...or heck, all 9 of them could have the same birthday.

 

If you think about it that way, then the problem is painful, because that's a lot of different ways to succeed, and each one is pretty hard to count!  But there's only one possibility left:  that nobody shares the same birthday.  If we can figure that out instead, then we know the probably of not having a match.  Not having a match, plus having a match, would have to add up to 100%, because those are the only options.  Either two people share a birthday (at least), or no two people share birthdays. 

 

So, now, the question is, how do we figure out if no two people share the same birthday?  Well, the first person walks in and has a birthday.  No one else is there yet, so he can't match anyone.

 

Second person walks in.  What is the probability that his birthday isn't a match with the first person's birthday?  That is, how many days out of the year could his birthday be and not be a match with the first person's birthday?

 

Third person walks in.  What is the probability that his birthday isn't a match with either of the other two people's birthdays?

 

And so on, until the 9th person. 

 

If you do it right, you end up with a bunch of different probabilities that all have to happen in order to get 9 unique birthdays in the room.  If you can figure out how to combine those probabilities, and then figure out what to do with that answer, you're there.

 

Let us know if that's enough to get you started!