Step by step
The question
Question: A survey asked boys and girls: tea or coffee?
Boys: 6 tea, 14 coffee.
Girls: 9 tea, 11 coffee.
One person is picked at random.
Find P(boy and tea) and P(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: 6 plus 14 is 20.
9 plus 11 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
Boy and tea: both at once.
That is one cell, 6.
Out of everyone, 40.
So P of boy and tea is 6/40.
Careful!
And means out of everyone: 40.
Over 15 would be given tea.
Both 6 and 40 can be divided by 2.
So it is 3/20.
3 A total over all
All the tea: 15.
Everyone: 40.
So P of tea is 15/40.
Both 15 and 40 can be divided by 5.
So it is 3/8.
So: P of boy and tea is 3/20.
P of tea is 3/8.
4 Check
Let's check.
The four cells add up to 40, everyone.
The table is right.
And the and rule: P of boy, 20/40, times P of tea given boy, 6/20.
The 20s cancel: 6/40.
So the answer is right!
Let's look back.
First: make the table.
Then: one cell over all.
Then: a total over all.
Last: check.