Step by step
The question
Question: A survey asked cats and dogs: indoor or outdoor?
Cats: 16 indoor, 4 outdoor.
Dogs: 10 indoor, 20 outdoor.
One person is picked at random.
Find P(indoor given dog).
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: 16 plus 4 is 20.
10 plus 20 is 30.
Column totals: 26 and 24.
And 50 people in all.
The totals sit at the edges, in the margins.
2 Keep only the given row
Given dog: only this row counts.
In it: 10 indoor, out of 30.
So P of indoor given dog is 10/30.
Careful!
Given dog: the bottom is only the dog row, 30.
Not all 50.
Both 10 and 30 can be divided by 10.
That makes it simpler.
So it is 1/3.
3 Check
Let's check.
The four cells add up to 50, everyone.
The table is right.
And the dog cells add up to 30: 10 plus 20.
So the bottom is right!
4 Pick the choice
Our answer is P(indoor | dog) = 1/3.
That is choice B.
5 Why not the others
A is 1/5.
Wrong: 10 out of all 50: that is P(dog and indoor).
That is the Careful mistake from before.
C is 5/13.
Wrong: 10 out of the 26 indoor pets: that is P(dog given indoor).
D is 2/3.
Wrong: 20 out of 30: that is outdoor, not indoor.
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.