You can know the number of games from "the game goes on until one person has two or six more games than the other"

The possible values are:

2 (Party A and Party B get 2 points and one party gets 0 points), the question has been given.

4 (3 points for Party A and Party B, 1 1). Since the outcome of each game is independent, P(ξ=4) means that the first two games stop, that is, 1-5/9, and the last two games stop, that is, 5/9, so the answer you see is (1-5/9) x5/9.

6 (Party A and Party B get 4 points, one gets 2 points, or everyone gets 3 points), which is easier to understand than your answer: p (ξ = 6) =1-p (ξ = 2)-p (ξ = 4) =1-5/.