I have a probability problem I'm trying to sort out, but it's been a long time since I've done this kind of stuff, so I'm hoping someone here can steer me towards some simple answers for the following scenario.

I have eight 8-sided dice in various colors. (The colors are irrelevant, but for the sake of this question, let's say they're red, yellow, blue, green, white, orange, purple and gray).

In each round of the game, I take each die and roll it once. The goal of the game is to eventually roll a 1 with each die. If I roll a 1, that is considered a "hit" and the die is removed from the game.

In each subsequent round, I roll all the remaining dice, again removing any that come up with a 1. Rounds continue in this manner until I've finally rolled a 1 with each die and there are no dice remaining.

What I'm trying to sort out is how to calculate the probability of whatever result of I can think of. What are the odds the game ends in the first round? Or the fourth round, or eighth round? What are the odds the game lasts longer than 10 rounds? What are the odds I go 12 rounds and still have 4 dice left?

I'm not so much looking for answers to the specific scenarios above, but the formula I'd use to calculate those odds. Help me, math wizards!