ShmoopTube

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Here's an unshmoopy question you'll find on an exam somewhere in life...

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At Washington High School, 32% of students take a music class.

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80% of students who take music also take an advanced math course.

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36% of the students in the math course do not take a music class.

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If a student does not take a music class,

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what is the probability that she takes advanced math?

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And here are the possible answers...

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Blah, blah, blah. So many percentages in this question...and they didn't bother to just

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calculate one more? Fine... WE'LL do it... The questions asks us: if a student does NOT

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take a music class, what's the probability that she takes advanced math?

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How should we write that in probability notation?

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We're looking for the probability that a

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student is in math GIVEN THAT she's not in music.

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We represent a "given statement" with

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a straight vertical bar...like this... Great, so now we've written the probability

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that she takes math given that she doesn't take music.

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Think back to the conditional probability formulas you should have memorized... the

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probability of B given A equals the probability of A and B divided by the probability of A.

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Translating this to the variables math and music....we have to find the probability of

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no music AND math... which is just the probability of taking ONLY math and dividing it by the

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probability of just math. All right, keep these in mind. We'll need

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these values in order to find what we want.

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You know those diagrams our teacher made us

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draw to analyze the similarities and differences between two things... Venn diagrams?

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Well, we can use a Venn diagram here to show the number of students taking math, the number

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of students taking music, and the overachievers who are taking both.

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Labeling the left side with math and the right side with music... the first thing we're given

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is that 32% of students take a music class. We can indicate this by labeling the entire

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music circle "32%." The next statement we're given is that 80%

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of students who take music also take an advanced math course. This is equivalent to saying

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that the probability of a student taking math GIVEN THAT they take music is 80%.

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Using the conditional probability rule, we can just multiply .8 times .32 to get the

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probability of students that take music AND math.

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So we have .8 times .32 is .256...or 25.6%.

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Finally, we're given that 36% of the students

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in the math course do not take a music class. So .36 times P(math) is the shaded left region

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of the math circle, not including the intersection of math and music.

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BUT we just solved for the intersection of math and music as .256...so we know the complement

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of .36 times the probability of math is .64 times the probability of math.

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Setting the two equal, we get that .64 times the probability of math equals .256.

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Divide both sides by .64, and the probability of taking math is... 40 percent.

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We figured out earlier that .36 times P(math) was the probability of students that ONLY

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take math...so now that we have that value,

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we can just multiply .36 times .4... to get .144. PHEW.

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Ok, now back to the formula we set in

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the very, very beginning,

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about oh... three hours ago... and plugging in our values...

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.144 divided by .68 equals around .212.

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The best option is choice (A), or 21.2%.

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And that is music... or math... to our ears.

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