Step by step
The question
Question: People were asked: bus or car?
18 juniors chose bus and 12 chose car.
10 seniors chose bus and 20 chose car.
One person who chose bus 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 bus: we only look at the people who chose bus.
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.
18 juniors chose bus, 12 chose car.
18 plus 12 is 30.
10 seniors chose bus, 20 chose car.
10 plus 20 is 30.
The totals: 28 chose bus, 32 chose car, 60 in all.
2 Keep only the given column
Given bus: only this column counts.
The others do not.
In it: 10 are seniors, out of 28.
3 Write the fraction
So the probability is 10/28.
Careful!
The bottom is only the people who chose bus: 28.
Not all 60.
4 Simplify
Both 10 and 28 can be divided by 2.
That makes it simpler.
So it is 5/14.
5 Check
Let's check with the rule: the probability of both, divided by the probability of bus.
Both: 10 out of all 60.
Bus: 28 out of 60.
Dividing by a fraction: flip it and multiply.
The 60s cancel: 10/28.
The same.
So the answer is right!
6 Pick the choice
Our answer is P(senior | bus) = 5/14.
That is choice C.
7 Why not the others
A is 1/3.
Wrong: that is P(bus given senior): the other way round.
B is 1/6.
Wrong: used all 60 students: only the 28 bus riders count.
That is the Careful mistake from before.
D is 9/14.
Wrong: that is a junior.
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.