For the birthday problem (Assuming even distribution across months), there are five people. What is the probability that three of the five are born in January and the other two are born in February?
I was going about the problem like this: I thought that three of the people born in January would be the same as "exactly 3 are born in January". Then I got 5 choose 3 ways multiplied by {1/ 12^3}
Then, for the second part there are 2 people left, and 11 months left to choose from. 2 choose 2 would be 1, and the probability that they would have the same birthday is 1/11^2.
I assumed that you would have to add these two to find the total. However, I think that I am on the wrong track to solving this question...and I don't know whether they should be added or multiplied.
Not really sure how to put these two parts together, please help.
You are correct in your method of counting here if you go on and multiply these two evaluations since here the second evaluation is preconditioned to each of the cases ( whatever they may be) in the first evaluation. Helps to think of it in counting total number of such cases and then dividing them by $12^5$