Let's say I have a fair 12-sided die. If I roll it 8 times, you can calculate the odds of rolling any single number, say, 4 or more times.
Running a simulation, I get that it's about p≈0.031
What, then, is the formula to calculate the probability of rolling a fair N-sided die M times and getting any single number X or more times?
This isn't a homework question. The specific example came up in a project I was working on, and I would like to have a formula for the general case instead of having to run simulations.