AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 29 Medium

Two players A and B compete for the championship in a Go match. The match follows a “best of three” format. The probability that A wins any individual game is \frac{2}{3}, and the outcomes of different games are independent. Given that A wins the championship, what is the probability that the match lasts for 3 games?

  • A.

    \frac{3}{10}

  • B.

    \frac{7}{10}

  • C.

    \frac{1}{2}

  • D.

    \frac{2}{5}

  • E.

    \frac{1}{5}

Answer:D

From the problem, the probability that A wins the championship is \frac{2}{3}\cdot\frac{2}{3}+\frac{2}{3}\cdot\frac{1}{3}\cdot\frac{2}{3}+\frac{1}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}=\frac{20}{27}.

The probability that the match lasts for 3 games is \frac{2}{3}\cdot\frac{1}{3}\cdot\frac{2}{3}+\frac{1}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}=\frac{8}{27}.

Therefore, given that A wins the championship, the probability that the match lasted for 3 games is \frac{\frac{8}{27}}{\frac{20}{27}}=\frac{2}{5}.

Hence, the answer is \frac{2}{5}.

Related Problems
Problem 26 AMC 10 Weekly Practice Round 3
Problem 27 AMC 10 Weekly Practice Round 3
Problem 28 AMC 10 Weekly Practice Round 3
Problem 30 AMC 10 Weekly Practice Round 3
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