# Series Winner Probability

Suppose we have Series 1 and Series 2, and four teams $$A,B,C,D$$. $$A$$ and $$B$$ play in Series 1 and $$C$$ and $$D$$ play in Series 2. In each series, the first team to win 3 games wins the series. The home team has a $$3/4$$ chance of winning in each game. The winner of each series plays each other in a similar format. What is the probability of $$A$$ defeating $$D$$ in the final series $$3-2$$?

The home team is based on:

Game #              Home Team
Game 1                 A/C
Game 2                 A/C
Game 3                 B/D
Game 4(if necessary)   B/D
Game 5(if necessary)   A/C

Can the probability be decomposed to something like: $$P(\text{A defeats D 3-2}) = P(\text{A defeats D 3-2}|\text{A wins Series 1 and D wins Series 2})$$