Step by step
The question
Question: People were asked: art or music?
12 juniors chose art and 18 chose music.
20 seniors chose art and 10 chose music.
One person who chose art is picked at random.
What is the probability they are one of the seniors?
There are four choices.
First we solve it, the normal way.
Then we pick.
The line means given.
Given they chose art: we only look at the people who chose art.
P stands for probability.
The line stands for given.
1 Make a table
One row for each group, one column for each answer, and the totals.
12 juniors chose art, 18 chose music.
12 plus 18 is 30.
20 seniors chose art, 10 chose music.
20 plus 10 is 30.
The totals: 32 chose art, 28 chose music, 60 in all.
2 Keep only the given column
Given art: only this column counts.
The others do not.
In it: 20 are seniors, out of 32.
3 Write the fraction
So the probability is 20/32.
Careful!
The bottom is only the people who chose art: 32.
Not all 60.
4 Simplify
Both 20 and 32 can be divided by 4.
That makes it simpler.
So it is 5/8.
5 Check
Let's check with the rule: the probability of both, divided by the probability of art.
Both: 20 out of all 60.
Art: 32 out of 60.
Dividing by a fraction: flip it and multiply.
The 60s cancel: 20/32.
The same.
So the answer is right!
6 Pick the choice
Our answer is P(senior | art) = 5/8.
That is choice C.
7 Why not the others
A is 1/3.
Wrong: 20 out of all 60 students: we only look at the 32 who chose art.
That is the Careful mistake from before.
B is 3/8.
Wrong: 12 out of 32: that is a junior.
D is 2/3.
Wrong: 20 out of 30 seniors: that is P(art given senior).
Let's look back.
First: make a table.
Then: keep only the given column.
Then: write the fraction.
Then: simplify.
Then: check.
Then: pick the choice.
Last: why not the others.