Step by step
The question
Question: A survey asked boys and girls: cat or dog?
Boys: 12 cat, 8 dog.
Girls: 10 cat, 20 dog.
One person is picked at random.
Find P(girl and dog) and P(dog given girl).
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: 12 plus 8 is 20.
10 plus 20 is 30.
Column totals: 22 and 28.
And 50 people in all.
The totals sit at the edges, in the margins.
2 One cell over all
Girl and dog: both at once.
That is one cell, 20.
Out of everyone, 50.
So P of girl and dog is 20/50.
Careful!
And means out of everyone: 50.
Over 28 would be given dog.
Both 20 and 50 can be divided by 10.
So it is 2/5.
3 Keep only the given row
Given girl: only this row counts.
In it: 20 dog, out of 30.
So P of dog given girl is 20/30.
Both 20 and 30 can be divided by 10.
So it is 2/3.
So: P of girl and dog is 2/5.
P of dog given girl is 2/3.
4 Check
Let's check.
The four cells add up to 50, everyone.
The table is right.
And the and rule: P of girl, 30/50, times P of dog given girl, 20/30.
The 30s cancel: 20/50.
So the answer is right!
Let's look back.
First: make the table.
Then: one cell over all.
Then: keep only the given row.
Last: check.