# Math Help - Facebook Friends' Birthdays: A Probability Question

1. ## Facebook Friends' Birthdays: A Probability Question

Hey guys,

I have 2,000 Facebook friends and noticed that not one of them has a birthday today (January 10th). I quickly calculated the probability of this as (364/365) ^ 2000 = 0.4% (assuming, for the purposes of this calculation, that each birthday is equally likely).

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?

I realize it would be 1-P(birthdays on all 365 days); that said, I have no idea how to calculate P(birthdays on all 365 days) either.

Please assume, for the purposes, of this calculation that there are no leap days and that each birthday is equally likely.

2. 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 $N=365$ (number of days) and $n=2000$ (number of friends). If $A_i$ denotes the event "day number $i$ is nobody's birthday", then we have:

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

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

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

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

3. better guestion, who would have 2000 'friends' at facebook?