AMC 8 Daily Practice Round 9

Complete problem set with solutions and individual problem pages

Problem 17 Medium

An opaque bag contains two balls, one red and one black, which are identical in size and shape. A ball is randomly drawn from the bag three times, with replacement. If drawing a red ball earns 2 points and drawing a black ball earns 1 point, what is the probability that the total score after three draws is exactly 5 points?

  • A.

    \frac{1}{4}

  • B.

    \frac{1}{2}

  • C.

    \frac{3}{8}

  • D.

    \frac{5}{8}

  • E.

    \frac{3}{4}

Answer:C

The possible total scores after 3 draws are 3, 4, 5, and 6, resulting in 4 different cases.

- The probability of scoring 3 points (drawing three black balls) is:     \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}

- The probability of scoring 6 points (drawing three red balls) is:     \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}

- Scoring 4 points requires 1 red ball and 2 black balls.

- Scoring 5 points requires 2 red balls and 1 black ball.

Since there are only two types of balls, each ball has an equal probability of being drawn (\frac{1}{2} per draw). The probabilities of scoring 4 and 5 must be the same.

Thus, the probability of scoring 5 points is:   \frac{1 - \frac{1}{8} - \frac{1}{8}}{2} = \frac{3}{8}

The answer is \text{C}.

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