Step by step
The question
Question: A survey asked boys and girls: art or music?
Boys: 9 art, 6 music.
Girls: 7 art, 8 music.
One person is picked at random.
Find P(music) and P(music 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: 9 plus 6 is 15.
7 plus 8 is 15.
Column totals: 16 and 14.
And 30 people in all.
The totals sit at the edges, in the margins.
2 A total over all
All the music: 14.
Everyone: 30.
So P of music is 14/30.
Careful!
The bottom is everyone, 30.
Not only the other 16.
Both 14 and 30 can be divided by 2.
So it is 7/15.
3 Keep only the given row
Given girl: only this row counts.
In it: 8 music, out of 15.
So P of music given girl is 8/15.
So: P of music is 7/15.
P of music given girl is 8/15.
4 Check
Let's check.
The four cells add up to 30, everyone.
The table is right.
And the two cells of music add up to its total: 6 plus 8 is 14.
So the answer is right!
Let's look back.
First: make the table.
Then: a total over all.
Then: keep only the given row.
Last: check.