Step by step
The question
Question: A survey asked men and women: yes or no?
Men: 16 yes, 4 no.
Women: 12 yes, 8 no.
One person is picked at random.
Find P(no given woman).
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: 16 plus 4 is 20.
12 plus 8 is 20.
Column totals: 28 and 12.
And 40 people in all.
The totals sit at the edges, in the margins.
2 Keep only the given row
Given woman: only this row counts.
In it: 8 no, out of 20.
So P of no given woman is 8/20.
Careful!
Given woman: the bottom is only the woman row, 20.
Not all 40.
Both 8 and 20 can be divided by 4.
That makes it simpler.
So it is 2/5.
3 Check
Let's check.
The four cells add up to 40, everyone.
The table is right.
And the rule: P of both, divided by P of woman.
Flip and multiply: the 40s cancel.
8/20.
So the answer is right!
Let's look back.
First: make the table.
Then: keep only the given row.
Last: check.