I am trying to find a way to calculate the chances of one player winning a game on badminton match to a specific score:
a Match of badminton is to the best of 3 games. Each game is won by the player who reaches 21 points first. But has to win by two. In case a 20-20 draw happens, then one player has to get two additional points. But this rule only applies until 30, where one player can win by 30-29. I hope this was clear.
So, I wanted to calculate the chances of a 21-0 score, 21-1, 21-2, etc until the 30-29.
I have done this assuming that the chances of Player A winning one point are the inverse of player B: So, if Pa is Players A chances of winning one point, 1-Pa=Pb chances of Player B winning one point.
But I see this approach as weak. In fact we should have 4 parameters:
Pa= player A chances of winning a point when serving Qb= player B chances of winning a point when receiving Pb= player B chances of winning a point when serving Qa= player A chances of winning a point when receiving.
So, now my problem is that I don't know how to calculate, for each possible scoreline, how many combinations we can get. And thing can get even more complicated when we assume that can happen points in a row or not:
For instance, ABAB is not the same as AABB, although this is 2-2 in terms of scoreline. Because on the ABAB both players lost their serve twice, and on the second example, both won one point on serve and lost one point on serve.
Can you point me on the right direction on how to solve this ?
thank you