Question

一个广播节目有一个由 $3$ 道选择题组成的测验，每题有 $3$ 个选项。参赛者答对 $2$ 题或更多即获胜。参赛者随机回答每一道题。获胜的概率是多少？

A radio program has a quiz consisting of $3$ multiple-choice questions, each with $3$ choices. A contestant wins if he or she gets $2$ or more of the questions right. The contestant answers randomly to each question. What is the probability of winning?

$\dfrac{1}{27}$

$\dfrac{1}{9}$

$\dfrac{2}{9}$

$\dfrac{7}{27}$

$\dfrac{1}{2}$

答案：D

知识点：二项概率分类讨论

难度评级：1140

解答：

全部 $3$ 题答对的概率为恰好答对 $2$ 题的概率为合计概率为 $\left(\dfrac13\right)^3=\dfrac1{27}.$ $\binom32\left(\dfrac13\right)^2\left(\dfrac23\right)=\dfrac6{27}.$ $\dfrac6{27}+\dfrac1{27}=\dfrac7{27}.$

所以正确答案是 D。

The probability that a contestant gets all $3$ correct is $\left(\dfrac13\right)^3=\dfrac1{27}.$ The probability of getting exactly $2$ correct is $\binom32\left(\dfrac13\right)^2\left(\dfrac23\right)=\dfrac6{27}.$ The combined probability is $\dfrac6{27}+\dfrac1{27}=\dfrac7{27}.$

Thus, the correct answer is D .

2017 AMC 10B 第 9 题

答案：D

难度评级：1140

解答：

其他年份的第 9 题