Efficient Method for Calculating the Probability of a Set of Outcomes?

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?

• Have you read the faq? I don't think this is a question of interest to research mathematicians. – Gerry Myerson Oct 3 '10 at 3:51
• you might try posting your question to stats.stackexchange.com they seem to deal with more basic stuff as well. [altho i'm not sure how "basic" you question actually is. i suppose with statistical software the answer for any particular outcome can be computed. maybe you want to ask about appropriate software for doing the required calculations. – ronaf Oct 3 '10 at 4:02
• Maybe you could edit the question so it asks for an efficient means of calculating the probability of each outcome (or a proof that there is no efficient way), if that's what you meant. – Bjørn Kjos-Hanssen Oct 3 '10 at 6:16
• @Bjørn: updated, thanks for the feedback! – Kenny Oct 3 '10 at 6:22