Formula for best of X?

If I'm trying to calculate the odds of winning a "Best of X rounds" game with probabilities other than 50/50, how would I go about that? I know there must be a relatively easy formula, rather than manual calculations...


Say I wanted to figure out the odds of winning a best of 5 game, where each round has a probability of 55% for me winning, and 45% of me losing. In this scenario, I have to win 3 rounds.

Is there a formula that would allow me to calculate that? I can't find it anywhere (maybe I don't know what to search for.)

In a best of 5, you have to add up the probabilities.

Probility of winning the first 3 = .55 ^ 3. = .166375
Probabilty of winning in four games = .55^3 * .45 * 3 = .224606 (3 ways to win in four games)
Probability of winning in five games = .55^3*.45^2 * 6 = .202146 (6 ways to win in five games)
Total = .593127

to check your work, add in the odds of

losing the first 3 = .45 ^ 3 = .091125
losing in 4 = .45^3 *.55 *3 = .150356
losing in 5 = .45^3 *.55^2*6 = .165392

Total = .406873

Ah, OK, I knew how to do it that way, I was just 1) brain farting last night and 2) looking for an easier way to do the math, so I could easily calculate best of 7, etc.

Anyway, thanks.