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Series from single-game odds

Market makingHard

A bookmaker quotes decimal odds of $1.5$ on team X and $3$ on team Y for a single game, with no margin. A series consists of $3$ independent games with the same probabilities, and the first team to win $2$ games takes the series. What is the probability that X wins the series? Give an exact fraction.

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Hint

Get the single-game probability from the odds, then count the ways to win the series.

Solution

The single-game probability of X is $\frac{1}{1.5} = \frac{2}{3}$ .
X wins the series if it wins at least $2$ of $3$ games: $3 \times (\frac{2}{3})^{2} \times (\frac{1}{3}) + (\frac{2}{3})^{3} = \frac{12}{27} + \frac{8}{27}$ .
The probability is $\frac{20}{27}$ .

Answer: $\frac{20}{27}$

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