AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 28 Medium

A medical device has two consumable components, A and B. After each use, the probability that component A needs to be replaced is 0.3, and the probability that component B needs to be replaced is 0.5. Given that at least one component must be replaced after the first use, what is the probability that both A and B need to be replaced?

  • A.

    0.15

  • B.

    0.65

  • C.

    \frac{3}{13}

  • D.

    \frac{5}{13}

  • E.

    \frac{6}{13}

Answer:C

Let event E be: at least one component needs to be replaced after the first use, and event F be: both components A and B need to be replaced.

Then,

P(E) = 1 - (1-0.3)(1-0.5) = 0.65,

P(EF) = 0.3 \times 0.5 = 0.15.

By the conditional probability formula,

P(F \mid E) = \frac{P(EF)}{P(E)} = \frac{0.15}{0.65} = \tfrac{3}{13}.

Therefore, the answer is C.

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