I wonder if someone can help me.

I'm looking to estimate the probability of player a v player b winning a darts match where the player who wins 6 legs first wins the match. I would normally use a negative binomial distribution for this kind of situation. However, darts is complicated because the player who throws first alternates between the players after each leg and this changes the probability of the leg outcome.

For example:

Leg 1: Player A throws first (Player A 63% win chance, Player B 37% win)

Leg 2: Player B throws first (Player A 45% win chance, Player B 55% chance)

Leg 3: Player A throws first (Player A 63% win chance, Player B 37% win)

Leg 4: Player B throws first (Player A 45% win chance, Player B 55% chance)

And so on. Any help would be hugely appreciated as I'm banging my head against a brick wall!