2021 AMC 10 A Fall

Complete problem set with solutions and individual problem pages

Problem 9 Easy

When a certain unfair die is rolled, an even number is 3 times as likely to appear as an odd number. The die is rolled twice. What is the probability that the sum of the numbers rolled is even?(2021 AMC Fall 10A, Question #9)

  • A.

    \frac{3}{8}

  • B.

    \frac{4}{9}

  • C.

    \frac{5}{9}

  • D.

    \frac{9}{16}

  • E.

    \frac{5}{8}

Answer:E

Since an even number is 3 times more likely to appear than an odd number, the probability of an even number appearing is \frac{3}{4}. Since the problem states that the sum of the two die must be even, the numbers must both be even or both be odd. We either have EE or OO, so we have \frac{3}{4} \cdot \frac{3}{4}+\frac{1}{4} \cdot \frac{1}{4}=\frac{1}{16}+\frac{9}{16}=\frac{10}{16}=(\mathbf{E}) \frac{5}{8}

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