Step by step
The question
Question: A survey asked walk and drive: yes or no?
Walk: 18 yes, 12 no.
Drive: 27 yes, 43 no.
One person is picked at random.
Find P(yes given walk).
There are four choices.
First we solve it, the normal way.
Then we pick.
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: 18 plus 12 is 30.
27 plus 43 is 70.
Column totals: 45 and 55.
And 100 people in all.
The totals sit at the edges, in the margins.
2 Keep only the given row
Given walk: only this row counts.
In it: 18 yes, out of 30.
So P of yes given walk is 18/30.
Careful!
Given walk: the bottom is only the walk row, 30.
Not all 100.
Both 18 and 30 can be divided by 6.
That makes it simpler.
So it is 3/5.
3 Check
Let's check.
The four cells add up to 100, everyone.
The table is right.
And the walk cells add up to 30: 18 plus 12.
So the bottom is right!
4 Pick the choice
Our answer is P(yes | walk) = 3/5.
That is choice C.
5 Why not the others
A is 9/50.
Wrong: 18 out of all 100: that is P(walk and yes).
That is the Careful mistake from before.
B is 2/5.
Wrong: 18 out of the 45 who said yes: that is P(walk given yes).
D is 9/10.
Wrong: 18 out of 20: 20 is not a total in the table.
Let's look back.
First: make the table.
Then: keep only the given row.
Then: check.
Then: pick the choice.
Last: why not the others.