What are the chances of my server giving an error given $N$ daily users? This probability question is based on a real problem: My server gives an error if it gets hit by more than $5$ requests in one second. If I have $N$ daily users, and each one sends an average of $M$ requests to the server per day (assuming each request takes exactly one second), what are the chances that the server will give an error that day?
Specifically, for an average day, I'd like to know $P(\geq 1 \space error)$ (the probability that the server gives at least one error that day), as well as $E[\#errors]$ (the expected value of the total number of errors that day)--so that I can  then calculate the expected number of errors over the course of one year, for example.

What I have so far: For any given user, whenever they send a request to the server, what are the chances another one of the $N-1$ users is doing it at the same time? There are $86,400$ seconds in one day, and each user is sending a request for $M$ of those seconds, so the chances are:
$$1 - \left(\frac{86,400-M}{86,400}\right)^{N-1}$$
Is that correct? If so, what are the chances that this happens to any user, not just a given $one$?

The probability that a specific user sends a request in a specific second is $p=\frac{M}{86,400}$. Then the number of requests received by the server in a specific second, noted $X$, follows a binomial law, $X\sim\mathcal{B}(N,p)$.
So $\mathbb{P}\{X\geq 5\}$ gives you the probability to have an error for a specific second. I'll let you compute this probability ; and then continue the computation for each second in the day.
