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(yes), P(woman and no) and 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 A total over all
All the yes: 28.
Everyone: 40.
So P of yes is 28/40.
Careful!
The bottom is everyone, 40.
Not only the other 12.
Both 28 and 40 can be divided by 4.
So it is 7/10.
3 One cell over all
Woman and no: both at once.
That is one cell, 8.
Out of everyone, 40.
So P of woman and no is 8/40.
Both 8 and 40 can be divided by 8.
So it is 1/5.
4 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.
Both 8 and 20 can be divided by 4.
So it is 2/5.
So: P of yes is 7/10.
P of woman and no is 1/5.
P of no given woman is 2/5.
5 Check
Let's check.
The four cells add up to 40, everyone.
The table is right.
And the two cells of yes add up to its total: 16 plus 12 is 28.
So the answer is right!
Let's look back.
First: make the table.
Then: a total over all.
Then: one cell over all.
Then: keep only the given row.
Last: check.