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Zero loss in mathematics competition
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/.