AMC 10 Weekly Practice Round 3

Complete problem set with solutions and individual problem pages

Problem 27 Medium

A and B each shoot once at the same target. The probability that A hits the target is \tfrac{1}{3}, and the probability that B hits the target is \tfrac{1}{2}. Given that the target is hit at least once, what is the probability that A hits the target?

  • A.

    \frac{1}{4}

  • B.

    \frac{1}{3}

  • C.

    \frac{1}{2}

  • D.

    \frac{2}{3}

  • E.

    \frac{3}{4}

Answer:C

Let event A be: the target is hit at least once, and event B be: A hits the target.

We calculate: P(A) = \frac{1}{3}\times\frac{1}{2} + \left(1-\frac{1}{3}\right)\times\frac{1}{2} + \frac{1}{3}\times\left(1-\frac{1}{2}\right) = \frac{2}{3},

P(AB) = \frac{1}{3}\times\frac{1}{2} + \frac{1}{3}\times\left(1-\frac{1}{2}\right) = \frac{1}{3}.

So, P(B|A) = \frac{P(AB)}{P(A)} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}.

Therefore, the answer is C.

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