Step by step
The question
Question: A survey asked juniors and seniors: bus or car?
Juniors: 14 bus, 6 car.
Seniors: 9 bus, 21 car.
One person is picked at random.
Find P(car given senior).
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Make the table
One row for each group, one column for each answer, and a total for each.
The four counts go inside.
Row totals: 14 plus 6 is 20.
9 plus 21 is 30.
Column totals: 23 and 27.
And 50 people in all.
The totals sit at the edges, in the margins.
2 Keep only the given row
Given senior: only this row counts.
In it: 21 car, out of 30.
So P of car given senior is 21/30.
Careful!
Given senior: the bottom is only the senior row, 30.
Not all 50.
Both 21 and 30 can be divided by 3.
That makes it simpler.
So it is 7/10.
3 Check
Let's check.
The four cells add up to 50, everyone.
The table is right.
And the senior cells add up to 30: 9 plus 21.
So the bottom is right!
4 Pick the choice
Our answer is P(car | senior) = 7/10.
That is choice C.
5 Why not the others
A is 3/10.
Wrong: 9 out of 30: that is the bus, not the car.
B is 21/50.
Wrong: 21 out of all 50: that is P(senior and car).
That is the Careful mistake from before.
D is 7/9.
Wrong: 21 out of the 27 car riders: that is P(senior given car).
Let's look back.
First: make the table.
Then: keep only the given row.
Then: check.
Then: pick the choice.
Last: why not the others.