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?

$\dfrac{1}{27}$

$\dfrac{1}{9}$

$\dfrac{2}{9}$

$\dfrac{7}{27}$

$\dfrac{1}{2}$

Answer: D

Concepts:binomial probability casework

Difficulty rating: 1140

Solution:

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 Problem 9

Answer: D

Difficulty rating: 1140

Solution:

Problem 9 in Other Years