2021 AMC 10 B Fall

Complete problem set with solutions and individual problem pages

Problem 14 Medium

Una rolls 6 standard 6 -sided dice simultaneously and calculates the product of the 6 numbers obtained. What is the probability that the product is divisible by 4 ?(2021 AMC Fall 10B, Question #14)

  • A.

    \frac{3}{4}

  • B.

    \frac{57}{64}

  • C.

    \frac{59}{64}

  • D.

    \frac{187}{192}

  • E.

    \frac{63}{64}

Answer:C

We will use complementary counting to find the probability that the product is not divisible by 4. Then, we can find the probability that we want by subtracting this from 1 . We split this into 2 cases.

Case 1: The product is not divisible by 2 . We need every number to be odd, and since the chance we roll an odd number is \frac{1}{2}, our probability is \left(\frac{1}{2}\right)^{6}=\frac{1}{64}.

Case 2 : The product is divisible by 2 , but not by 4 . We need 5 numbers to be odd, and one to be divisible by 2 , but not by 4 . There is a \frac{1}{2} chance that an odd number is rolled, a \frac{1}{3} chance that we roll a number satisfying the second condition (only 2 and 6 work), and 6 ways to choose the order in which the even number appears. Our probability is \left(\frac{1}{2}\right)^{5}\left(\frac{1}{3}\right) \cdot 6=\frac{1}{16}. Therefore, the probability the product is not divisible by 4 is \frac{1}{64}+\frac{1}{16}=\frac{5}{64}. Our answer is 1-\frac{5}{64}= (C) \frac{59}{64}

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