Step by step
The question
Question: A survey asked adults and children: tea or coffee?
Adults: 30 tea, 10 coffee.
Children: 5 tea, 15 coffee.
One person is picked at random.
Find P(coffee) and P(child given coffee).
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: 30 plus 10 is 40.
5 plus 15 is 20.
Column totals: 35 and 25.
And 60 people in all.
The totals sit at the edges, in the margins.
2 A total over all
All the coffee: 25.
Everyone: 60.
So P of coffee is 25/60.
Careful!
The bottom is everyone, 60.
Not only the other 35.
Both 25 and 60 can be divided by 5.
So it is 5/12.
3 Keep only the given column
Given coffee: only this column counts.
In it: 15 child, out of 25.
So P of child given coffee is 15/25.
Both 15 and 25 can be divided by 5.
So it is 3/5.
So: P of coffee is 5/12.
P of child given coffee is 3/5.
4 Check
Let's check.
The four cells add up to 60, everyone.
The table is right.
And the two cells of coffee add up to its total: 10 plus 15 is 25.
So the answer is right!
Let's look back.
First: make the table.
Then: a total over all.
Then: keep only the given column.
Last: check.