How many people must be in a room so that the probability of two of them being born in the same month is 0.9?

Answer 1

The easy way to do this is to calculate the opposite probability -- the probability that everyone is born in a different month. Then compute the point at which that number gets to less than or equal to 0.1, because the probability that a birth month is shared is 1.0 minus that number.

With two people, the probability is 11/12 that they are not both born in the same month. (The first guy can have any birthday at all, and the second guy's must fall in one of the 11 remaining months.) That's about 0.92, meaning the probability that they share the same birth month is only 0.08.

With three people it is 11/12 * 10/12. (The first guy can have any birthday at all, the second guy's must fall in one of the 11 remaining months, and the third guy's must fall in one of the remaining 10 months matching neither of the first two.) That's about 0.76, meaning it's only 0.24 that a birthdate is shared.

Add a fourth person, and continue the pattern by multiplying 11/12 * 10/12 * 9/12. It's down to 0.57 for four separate months, meaning (1 -0.57) = 0.43 for a shared birth month.

Add a fifth, and multiply by 8/12, to get 0.38. Now we're up to 0.62 for a shared birth month.

Add a sixth, and multiply by 7/12, to get 0.22. 0.78 for a shared birth month.

Add a seventh, multiply by 6/12, to get 0.11. 0.89 for a shared birth month.

Add an eighth, multiply by 5/12 to get 0.05. 0.95 for a shared birth month.

Depending on how you round off, the answer that you want is either seven people (0.89) or eight (0.95). Seven people is closer to 0.9, and rounds off to 0.9, but hasn't quite reached it. If they had asked "the minimum number of people where the probability of a shared birth month EXCEEDS 0.9" I'd go with eight. The way you worded the question, I'd answer seven.

Note that I am calculating "at least" one matching month, not "exactly" one matching month, which as far as I know never reaches 0.9.

Answer 2

Use the formula nCrP^rQ^n-r and so it would be xC2(.9)^2(.1)^x-2 and then i guess i would solve for that. It should work out if not im sorry.