I was assigned some optional practice on the birthday question. I got all the questions on the birthday problem itself correct, but although I got an answer for this last one (n >= 612.257), it seems VERY unintuitive to me. Can you please confirm whether it's right or what I did wrong? Here is the problem:
Given n people randomly chosen, find the smallest value of n so that the probability at least one of them were born on the same day of the year that you are born is at least 50%.
Now, here's what I thought:
P(X >= 1) >= 0.5
P(X >= 1) = 1 - P(X=0) - P(X=1)
1 - P(X=0) - P(X=1) >= 0.5
0.5 >= P(X=0) + P(X=1)
0.5 >= (nCr(n,0) * (1/365)^0 * (364/365)^n) + (nCr(n,1) * (1/365) * (364/365)^(n-1))
Now, nCr(anything, 0) is always 1, and nCr(n, 1) is always n. So:
0.5 >= (364/365)^n + (n/365)*(364/365)^(n-1)
Plugging this into wolfram alpha, I get that n >= 612.257. This result seems very unintuitive to me. Is it correct?