AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 30 Medium

Andy can choose to join from 2 science clubs, 4 art clubs, and 3 sports clubs. If Andy signs up for two clubs, then given that exactly one of them is an art club, what is the probability that the other one is a sports club?

  • A.

    \frac{3}{5}

  • B.

    \frac{6}{13}

  • C.

    \frac{1}{2}

  • D.

    \frac{3}{4}

  • E.

    \frac{2}{5}

Answer:A

Let event A be “one is an art club,” and event B be “one is a sports club,”

then

P(A)=\frac{_{4}C_{1}\times _{5}C_{1}}{_{9}C_{2}}=\frac{20}{36}=\frac{5}{9},

P(AB)=\frac{_{4}C_{1}\times _{3}C_{1}}{_{9}C_{2}}=\frac{12}{36}=\frac{1}{3}.

Thus, P(B|A)=\frac{P(AB)}{P(A)}=\frac{\frac{1}{3}}{\frac{5}{9}}=\frac{3}{5}.

Therefore, the answer is \text{A}.

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