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 of the seniors is picked at random.
What is the probability they chose bus?
There are four choices.
First we solve it, the normal way.
Then we pick.
The line means given.
Given one of the seniors: we only look at the seniors.
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 row
Given senior: only this row counts.
The others do not.
In it: 10 chose bus, out of 30.
3 Write the fraction
So the probability is 10/30.
Careful!
The bottom is only the seniors: 30.
Not all 60.
4 Simplify
Both 10 and 30 can be divided by 10.
That makes it simpler.
So it is 1/3.
5 Check
Let's check with the rule: the probability of both, divided by the probability of senior.
Both: 10 out of all 60.
Senior: 30 out of 60.
Dividing by a fraction: flip it and multiply.
The 60s cancel: 10/30.
The same.
So the answer is right!
6 Pick the choice
Our answer is P(bus | senior) = 1/3.
That is choice A.
7 Why not the others
B is 1/6.
Wrong: used all 60 students: only the 30 seniors count.
That is the Careful mistake from before.
C is 5/14.
Wrong: that is P(senior given bus): the other way round.
D is 2/3.
Wrong: that is by car.
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.