Step by step
The question
Question: A survey asked juniors and seniors: bus or car?
Juniors: 8 bus, 12 car.
Seniors: 6 bus, 14 car.
One person is picked at random.
Find P(junior and bus), P(junior or bus) and P(bus given junior).
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: 8 plus 12 is 20.
6 plus 14 is 20.
Column totals: 14 and 26.
And 40 people in all.
The totals sit at the edges, in the margins.
2 One cell over all
Junior and bus: both at once.
That is one cell, 8.
Out of everyone, 40.
So P of junior and bus is 8/40.
Careful!
And means out of everyone: 40.
Over 14 would be given bus.
Both 8 and 40 can be divided by 8.
So it is 1/5.
3 Add, take away the overlap
Junior: 20.
Bus: 14.
But 8 are both.
They are in both totals.
The or rule: add the two, then take away the overlap, so it is counted once.
Same bottoms: 20 plus 14 minus 8 is 26.
So it is 26/40.
Both 26 and 40 can be divided by 2.
So it is 13/20.
4 Keep only the given row
Given junior: only this row counts.
In it: 8 bus, out of 20.
So P of bus given junior is 8/20.
Both 8 and 20 can be divided by 4.
So it is 2/5.
So: P of junior and bus is 1/5.
P of junior or bus is 13/20.
P of bus given junior is 2/5.
5 Check
Let's check.
The four cells add up to 40, everyone.
The table is right.
And the and rule: P of junior, 20/40, times P of bus given junior, 8/20.
The 20s cancel: 8/40.
So the answer is right!
Let's look back.
First: make the table.
Then: one cell over all.
Then: add, take away the overlap.
Then: keep only the given row.
Last: check.