2017 AMC 10B Problem 9

Question

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}

Count exactly three correct plus exactly two correct

For exactly two correct, choose which question is missed

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 .

Problem 9 in other years