Let's say I'm playing 10 different independent "games". For each game, I know the probability of winning, the probability of tying, and the probability of losing.

From these values, I've also calculated the probability of winning exactly X games, the probability of tying exactly X games, and the probability of losing exactly X games (for X = 0 to 10).

I'm just trying to figure out the probability of each outcome after playing all 10 games. For example, what is the probability of winning 7, losing 2, and tying 1?

Any ideas? Thanks in advance!

Let's say I'm playing N different independent "games". For each game, I know the probability of winning, the probability of tying, and the probability of losing.

From these values, I've also calculated the probability of winning exactly X games, the probability of tying exactly X games, and the probability of losing exactly X games (for X = 0 to N).

I'm just trying to figure out the probability of each outcome after playing all N games. For example, if N = 10, what is the probability of winning 7, losing 2, and tying 1?

Any ideas, or a proof that this is impossible to compute efficiently?