AP Statistics 1.2 Anticipating Patterns
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?
|AP Statistics||Anticipating Patterns|
|Statistics and Probability||Make inferences and justify conclusions|
Make inferences and justify conclusions from sample surveys, experiments, and observational studies
what is the probability that she takes advanced math?
And here are the possible answers...
Blah, blah, blah. So many percentages in this question...and they didn't bother to just
calculate one more? Fine... WE'LL do it... The questions asks us: if a student does NOT
take a music class, what's the probability that she takes advanced math?
How should we write that in probability notation?
We're looking for the probability that a
student is in math GIVEN THAT she's not in music.
We represent a "given statement" with
a straight vertical bar...like this... Great, so now we've written the probability
that she takes math given that she doesn't take music.
Think back to the conditional probability formulas you should have memorized... the
probability of B given A equals the probability of A and B divided by the probability of A.
Translating this to the variables math and music....we have to find the probability of
no music AND math... which is just the probability of taking ONLY math and dividing it by the
probability of just math. All right, keep these in mind. We'll need
these values in order to find what we want.
You know those diagrams our teacher made us
draw to analyze the similarities and differences between two things... Venn diagrams?
Well, we can use a Venn diagram here to show the number of students taking math, the number
of students taking music, and the overachievers who are taking both.
Labeling the left side with math and the right side with music... the first thing we're given
is that 32% of students take a music class. We can indicate this by labeling the entire
music circle "32%." The next statement we're given is that 80%
of students who take music also take an advanced math course. This is equivalent to saying
that the probability of a student taking math GIVEN THAT they take music is 80%.
Using the conditional probability rule, we can just multiply .8 times .32 to get the
probability of students that take music AND math.
So we have .8 times .32 is .256...or 25.6%.
Finally, we're given that 36% of the students
in the math course do not take a music class. So .36 times P(math) is the shaded left region
of the math circle, not including the intersection of math and music.
BUT we just solved for the intersection of math and music as .256...so we know the complement
of .36 times the probability of math is .64 times the probability of math.
Setting the two equal, we get that .64 times the probability of math equals .256.
Divide both sides by .64, and the probability of taking math is... 40 percent.
We figured out earlier that .36 times P(math) was the probability of students that ONLY
take math...so now that we have that value,
we can just multiply .36 times .4... to get .144. PHEW.
Ok, now back to the formula we set in
the very, very beginning,
about oh... three hours ago... and plugging in our values...
.144 divided by .68 equals around .212.
The best option is choice (A), or 21.2%.
And that is music... or math... to our ears.