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Leading a long series

ProbabilityMedium

A series ends as soon as one team has $4$ wins. Each game is won by either team with probability $\frac{1}{2}$ , independently, with no draws. Team Red leads by $2$ wins to $1$ . What is the probability that Red wins the series? Give a fraction a/b in lowest terms.

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Hint

Pretend that all remaining games are played regardless; the winner of the series does not change.

Solution

Red needs $2$ more wins, and the opponent needs $3$ more wins.
The series is decided within at most $4$ further games, so imagine that exactly $4$ games are played.
Red wins the series exactly when it wins at least $2$ of these $4$ games.
The probability of at most $1$ win is $\frac{1 + 4}{16} = \frac{5}{16}$ .
So Red wins with probability $1 - \frac{5}{16} = \frac{11}{16}$ .
The answer is $\frac{11}{16}$ .

Answer: $\frac{11}{16}$

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