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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: $$P(\text{A defeats D 3-2}) = P(\text{A defeats D 3-2}|\text{A wins Series 1 and D wins Series 2})$$

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