AMC 8 2016 Problem 13

AMC8 2016 年第 13 题

概率★★★☆☆

Two different numbers are randomly selected from the set

\{ - 2, -1, 0, 3, 4, 5\}

and multiplied together. What is the probability that the product is 0 ?

从集合

\{-2, -1, 0, 3, 4, 5\}

中随机选取两个不同的数相乘。那么乘积为 0 的概率是多少？

A. $\dfrac{1}{6}$
B. $\dfrac{1}{5}$
C. $\dfrac{1}{4}$
D. $\dfrac{1}{3}$
E. $\dfrac{1}{2}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

1. Identify the total number of ways to select two different numbers from the set: The set has 6 elements. The number of ways to choose 2 different numbers from 6 is given by the combination formula:

\displaystyle \binom{6}{2} = \frac{6×5}{2×1} = 15

. 2. Identify the favorable outcomes: For the product to be zero, one of the chosen numbers must be zero. The set contains one zero (0). To have a product of zero, we need to choose 0 and any other number from the remaining five numbers -2, -1, 3, 4, 5. The number of ways to choose 0 and one other number from the remaining five is 5. 3. Calculate the probability: The probability is the number of favorable outcomes divided by the total number of outcomes:

\displaystyle \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{15} = \frac{1}{3}

. Thus, the probability that the product is 0 is

\displaystyle \frac{1}{3}

. Because the only way the product of the two numbers is 0 is if one of the numbers we choose is 0, we calculate the probability of NOT choosing a $0$. We get

\displaystyle \frac{5}{6} \cdot \frac{4}{5} = \frac{2}{3}

. Therefore our answer is

\displaystyle 1 - \frac{2}{3} = \frac{1}{3}

\displaystyle \frac{1}{6}

* Any other number + Any other number *

\displaystyle \frac{1}{6}

\displaystyle \frac{1}{6}+\frac{1}{6} = \frac{1}{3}

-JasonDaGoat

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