2017 AMC 10B 第 9 题

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}

分别计算恰好答对三题和恰好答对两题。

Count exactly three correct plus exactly two correct

恰好答对两题时，先选择哪一题答错。

For exactly two correct, choose which question is missed

全部

3

题答对的概率为

\left(\dfrac13\right)^3=\dfrac1{27}\text{。}

恰好答对

2

题的概率为

\binom32\left(\dfrac13\right)^2\left(\dfrac23\right)=\dfrac6{27}\text{。}

合计概率为

\dfrac6{27}+\dfrac1{27}=\dfrac7{27}\text{。}

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 .

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