Step by step
The question
Question: People were asked: bus or car?
15 freshmen chose bus and 25 chose car.
18 sophomores chose bus and 12 chose car.
One of the sophomores is picked at random.
What is the probability they chose car?
There are four choices.
First we solve it, the normal way.
Then we pick.
The line means given.
Given one of the sophomores: we only look at the sophomores.
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.
15 freshmen chose bus, 25 chose car.
15 plus 25 is 40.
18 sophomores chose bus, 12 chose car.
18 plus 12 is 30.
The totals: 33 chose bus, 37 chose car, 70 in all.
2 Keep only the given row
Given sophomore: only this row counts.
The others do not.
In it: 12 chose car, out of 30.
3 Write the fraction
So the probability is 12/30.
Careful!
The bottom is only the sophomores: 30.
Not all 70.
4 Simplify
Both 12 and 30 can be divided by 6.
That makes it simpler.
So it is 2/5.
5 Check
Let's check with the rule: the probability of both, divided by the probability of sophomore.
Both: 12 out of all 70.
Sophomore: 30 out of 70.
Dividing by a fraction: flip it and multiply.
The 70s cancel: 12/30.
The same.
So the answer is right!
6 Pick the choice
Our answer is P(car | sophomore) = 2/5.
That is choice C.
7 Why not the others
A is 6/35.
Wrong: 12 out of all 70 students: we only look at the 30 sophomores.
That is the Careful mistake from before.
B is 12/37.
Wrong: 12 out of the 37 car riders: that is P(sophomore given car).
D is 3/5.
Wrong: 18 out of 30: that is the bus.
Let's look back.
First: make a table.
Then: keep only the given row.
Then: write the fraction.
Then: simplify.
Then: check.
Then: pick the choice.
Last: why not the others.