Originally Posted by

**probabilitydunce** Here's my question, though: how would I calculate the probability of nobody (out of my 2,000 friends) having a birthday on at least one of the 365 calendar days? In other words, what are the odds that I don't have a birthday on each of the 365 days in the year?

Hi,

I think there is no simple formula for that one. Let $\displaystyle N=365$ (number of days) and $\displaystyle n=2000$ (number of friends). If $\displaystyle A_i$ denotes the event "day number $\displaystyle i$ is nobody's birthday", then we have:

$\displaystyle \{\text{not everyday is a birthday}\}=A_1\cup\cdots\cup A_N$.

Therefore, using inclusion-exclusion formula and the fact that $\displaystyle P(A_1\cap\cdots\cap A_k)=\left(\frac{N-k}{N}\right)^n$, we get:

$\displaystyle P(\text{not everyday is a birthday})=\sum_{k=1}^N (-1)^{k+1}{N\choose k}\left(\frac{N-k}{N}\right)^n$.

In your case, I think the probability is approximately 0.783881.