This is an AOPS question. This presentation is an adaptation of Lancelot Link’s solution for a closely related question.
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This is question has a subtle contingency and requires two probability equations to solve.
For the Giants to win the series in game six, they need to win three games in five trials. This is a binomial distribution.
P(G win series in 6 games)=P(G wins 3 in 5)∗P(G wins 6th game)=(53)(0.5)3(0.5)2∗(0.5)=0.15625
Further ... For the series to end in six games, either the Giants or the Royals must win the series in six games.
P(Series ends on game 6)=P(G wins series on game 6)+P(R wins series on game 6)=2P(G wins series on game 6)=(2)(53)(0.5)3(0.5)2∗(0.5)=516
The probability the Giants win the series in game 6 is 5/16.
GA
Edit: changed ASOP (the fabulist) to AOPS, (the fabled academy of higher mathematics).