Maria and Oleh play a match that ends when one of them has won 2 games. Maria wins each game with probability 2/3, independently of other games, with no draws. What is the probability that Maria wins the match? Give a fraction a/b in lowest terms.

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Maria and Oleh play a match that ends when one of them has won $2$ games. Maria wins each game with probability $\frac{2}{3}$ , independently of other games, with no draws. What is the probability that Maria wins the match? Give a fraction a/b in lowest terms.

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Hint

Split by the number of games played: $2$ or $3$ .

Solution

Maria wins in $2$ games with probability $(\frac{2}{3})^{2} = \frac{4}{9}$ .
Maria wins in exactly $3$ games if she wins the last game and exactly one of the first $2$ , which has probability $2 \times (\frac{2}{3})^{2} \times (\frac{1}{3}) = \frac{8}{27}$ .
Add the cases: $\frac{4}{9} + \frac{8}{27} = \frac{12}{27} + \frac{8}{27}$ .
This equals $\frac{20}{27}$ .
The answer is $\frac{20}{27}$ .

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

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