Formula to calculate the probability of rolling an N-sided die M times and getting X or more of any single number? [closed]

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.

• Hint: the binomial distribution. Dec 7, 2021 at 8:51
• Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. Dec 7, 2021 at 19:40
• Does this answer your question? Probability of throwing n different numbers in m throws of a dice
– Carl
Dec 9, 2021 at 19:49