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AP Statistics 1.2 Anticipating Patterns 349 Views


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AP Statistics: Anticipating Patterns Drill 1, Problem 2. If a student does not take a music class, what is the probability that she takes advanced math?

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Transcript

00:03

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

00:11

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

00:15

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

00:20

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

00:24

If a student does not take a music class,

00:27

what is the probability that she takes advanced math?

00:30

And here are the possible answers...

00:36

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

00:41

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

00:47

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

00:52

How should we write that in probability notation?

00:55

We're looking for the probability that a

00:57

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

01:01

We represent a "given statement" with

01:03

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

01:09

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

01:12

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

01:16

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

01:24

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

01:28

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

01:34

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

01:38

these values in order to find what we want.

01:44

You know those diagrams our teacher made us

01:45

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

01:52

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

01:57

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

02:03

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

02:07

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

02:12

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

02:17

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

02:22

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

02:29

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

02:34

probability of students that take music AND math.

02:37

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

02:43

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

02:46

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

02:53

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

02:58

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

03:03

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

03:11

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

03:16

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

03:22

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

03:27

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

03:31

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

03:36

Ok, now back to the formula we set in

03:39

the very, very beginning,

03:44

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

03:47

.144 divided by .68 equals around .212.

03:52

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

03:56

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

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