Multiple Events and Cumulative Probability Calculation(s)

#1
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.

Thanks!