Question

At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it?

$10\%$

$12\%$

$20\%$

$25\%$

$33\tfrac{1}{3}\%$

Answer: D

Concepts:conditional probability Bayes’ Theorem

Difficulty rating: 1440

Solution:

Students who like dancing but say they dislike it make up $60\% \cdot 20\% = 12\%$ of all students. Students who dislike dancing and say so make up $40\% \cdot 90\% = 36\%.$ Among everyone who says they dislike dancing, the fraction who actually like it is $\frac{12}{12 + 36} = \frac{12}{48} = \frac{1}{4} = 25\%.$

Thus, the correct answer is D.

2017 AMC 12B Problem 10

Answer: D

Difficulty rating: 1440

Solution:

Problem 10 in Other Years