Step by step
The question
Question: A survey asked boys and girls: yes or no?
Boys: 5 yes, 15 no.
Girls: 10 yes, 10 no.
One person is picked at random.
Find P(girl and yes).
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: 5 plus 15 is 20.
10 plus 10 is 20.
Column totals: 15 and 25.
And 40 people in all.
The totals sit at the edges, in the margins.
2 One cell over all
Girl and yes: both at once.
That is one cell, 10.
Out of everyone, 40.
So P of girl and yes is 10/40.
Careful!
And means out of everyone: 40.
Over 15 would be given yes.
Both 10 and 40 can be divided by 10.
That makes it simpler.
So it is 1/4.
3 Check
Let's check.
The four cells add up to 40, everyone.
The table is right.
And the and rule: P of girl, 20/40, times P of yes given girl, 10/20.
The 20s cancel: 10/40.
So the answer is right!
Let's look back.
First: make the table.
Then: one cell over all.
Last: check.