Step by step
The question
Question: A survey asked boys and girls: bike or bus?
Boys: 15 bike, 10 bus.
Girls: 12 bike, 13 bus.
One person is picked at random.
Find P(girl and bike) and P(girl given bike).
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: 15 plus 10 is 25.
12 plus 13 is 25.
Column totals: 27 and 23.
And 50 people in all.
The totals sit at the edges, in the margins.
2 One cell over all
Girl and bike: both at once.
That is one cell, 12.
Out of everyone, 50.
So P of girl and bike is 12/50.
Careful!
And means out of everyone: 50.
Over 27 would be given bike.
Both 12 and 50 can be divided by 2.
So it is 6/25.
3 Keep only the given column
Given bike: only this column counts.
In it: 12 girl, out of 27.
So P of girl given bike is 12/27.
Both 12 and 27 can be divided by 3.
So it is 4/9.
So: P of girl and bike is 6/25.
P of girl given bike is 4/9.
4 Check
Let's check.
The four cells add up to 50, everyone.
The table is right.
And the and rule: P of girl, 25/50, times P of bike given girl, 12/25.
The 25s cancel: 12/50.
So the answer is right!
Let's look back.
First: make the table.
Then: one cell over all.
Then: keep only the given column.
Last: check.