Multiple Events and Cumulative Probability Calculation(s)

Hi! I don't know statistics, hopefully someone will kindly help me.

I'd like to learn how to calculate the probability of an event happening Y times if it has an X chance of occurring, so that I can calculate how many attempts at X chance are required to provide an 80% and 95% probability of it occurring Y times.

Each time the event occurs is independent of all other events.

That might have been a bit wordy, let me try again:
If I have a 10-sided die, I would like to be able to calculate how many times I need to roll it in order to have a 90% probability of "rolling a 1" 4 times. They do not need to be sequential. Each roll is clearly independent of other rolls.

I'm interested in both calculating this as well as learning the fun maths behind it.