Birthday problem with an arbitrary "year" and number of overlapping birthdays

Question

First off, I'm not a mathematician, so if I'm going about this in the wrong way, please let me know.

The problem

I have ~4.1 million timestamps (with second accuracy) when people logged into a website. They were collected over a period of ~50.5 million seconds. Some of these records are erroneous (one says that 7044 people logged in at the same second). Assuming an even distribution of logins, find n where the probability of n people logging in simultaneously < 50%

Current approach

This seems like the birthday problem could be adapted here. Given a "year" of 50.5M days, and 4.1M people the chance that two people will not share a birthday is negligible.

I found this post with a formula for triplets (three simultaneous birthdays), but is there a way to generalize it for n simultaneous birthdays? I'm guessing it's not as simple as changing the 3 in the equation to an n.

Answer 1

Suppose you had $N$ possible birthdays and $M$ people (with birthdays chosen independently and uniformly). The probability of $k$ people sharing a birthday seems difficult to calculate, but much easier is the expected number of (unordered) $k$-tuples sharing a birthday. Namely, there are ${M \choose k}$ possible $k$-tuples, the probability that a particular $k$-tuple share the same birthday is $N^{1-k}$, so the expected number is ${M \choose k} N^{1-k}$. The expected number of $k$-tuples is an upper bound on the probability of existence of such a $k$-tuple.

If $M$ is very large, ${M \choose k} \approx M^k /k!$ so the expected number of $k$-tuples is approximately $M^k N^{1-k}/k!$. Thus in your case with $M = 4.1 \times 10^6$, $N = 5.05 \times 10^7$ and $k = 7044$, the expected number is ridiculously small, about $9.5 \times 10^{-31722}$. Even with $k = 6$, the expected number is about $0.020$. On the other hand, for $k = 5$ the expected number is about $1.484$. So it seems likely that the maximum number of people sharing a birthday is $5$.