AMC 8 2001 Problem 18

AMC8 2001 年第 18 题

概率★★★☆☆

Two dice are thrown. What is the probability that the product of the two numbers is a multiple of 5?

掷两枚骰子，问得到的两个数之积为 5 的倍数的概率是多少？

A. $\dfrac{1}{36}$
B. $\dfrac{1}{18}$
C. $\dfrac{1}{6}$
D. $\dfrac{11}{36}$
E. $\dfrac{1}{3}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

This is equivalent to asking for the probability that at least one of the numbers is a multiple of 5, since if one of the numbers is a multiple of 5, then the product with it and another integer is also a multiple of 5, and if a number is a multiple of 5, then since 5 is prime, one of the factors must also have a factor of 5, and 5 is the only multiple of 5 on a die, so one of the numbers rolled must be a 5. To find the probability of rolling at least one 5, we can find the probability of not rolling a 5 and subtract that from 1, since you either roll a 5 or not roll a 5. The probability of not rolling a 5 on either dice is

\displaystyle \left( \frac{5}{6} \right) \left( \frac{5}{6} \right)= \frac{25}{36}

. Therefore, the probability of rolling at least one five, and thus rolling two numbers whose product is a multiple of 5, is

\displaystyle 1 - \frac{25}{36} = \frac{11}{36}

\text{D}

The only way to get a multiple of 5 is to have at least one 5. If the first dice rolls a 5, there are 6 ways to get a multiple of 5. If the second dice rolls a 5, there are also 6 ways. However, one case is repeated: both dice roll a 5. Therefore, there are 6 + 6 - 1 = 11, and there is a total of 6×6 ways, so the probability is

\text{(D)}\ \dfrac{11}{36}

Solution by IHateMath314265169

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