Step by step
The question
Question: A survey asked adults and children: tea or coffee?
Adults: 30 tea, 10 coffee.
Children: 5 tea, 15 coffee.
One person is picked at random.
Find P(tea given adult) and P(adult given tea).
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: 30 plus 10 is 40.
5 plus 15 is 20.
Column totals: 35 and 25.
And 60 people in all.
The totals sit at the edges, in the margins.
2 Keep only the given row
Given adult: only this row counts.
In it: 30 tea, out of 40.
So P of tea given adult is 30/40.
Careful!
Given adult: the bottom is only the adult row, 40.
Not all 60.
Both 30 and 40 can be divided by 10.
So it is 3/4.
3 Keep only the given column
Given tea: only this column counts.
In it: 30 adult, out of 35.
So P of adult given tea is 30/35.
Both 30 and 35 can be divided by 5.
So it is 6/7.
So: P of tea given adult is 3/4.
P of adult given tea is 6/7.
4 Check
Let's check.
The four cells add up to 60, everyone.
The table is right.
And the rule: P of both, divided by P of adult.
Flip and multiply: the 60s cancel.
30/40.
So the answer is right!
Let's look back.
First: make the table.
Then: keep only the given row.
Then: keep only the given column.
Last: check.