2017 AMC 12B Problem 10

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?

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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.

Tagged: conditional probability · Bayes’ Theorem