Step by step
The question
Question: People were asked: cat or dog?
12 boys chose cat and 8 chose dog.
10 girls chose cat and 20 chose dog.
One person who chose dog is picked at random.
What is the probability they are one of the girls?
There are four choices.
First we solve it, the normal way.
Then we pick.
The line means given.
Given they chose dog: we only look at the people who chose dog.
P stands for probability.
The line stands for given.
1 Make a table
One row for each group, one column for each answer, and the totals.
12 boys chose cat, 8 chose dog.
12 plus 8 is 20.
10 girls chose cat, 20 chose dog.
10 plus 20 is 30.
The totals: 22 chose cat, 28 chose dog, 50 in all.
2 Keep only the given column
Given dog: only this column counts.
The others do not.
In it: 20 are girls, out of 28.
3 Write the fraction
So the probability is 20/28.
Careful!
The bottom is only the people who chose dog: 28.
Not all 50.
4 Simplify
Both 20 and 28 can be divided by 4.
That makes it simpler.
So it is 5/7.
5 Check
Let's check with the rule: the probability of both, divided by the probability of dog.
Both: 20 out of all 50.
Dog: 28 out of 50.
Dividing by a fraction: flip it and multiply.
The 50s cancel: 20/28.
The same.
So the answer is right!
6 Pick the choice
Our answer is P(girl | dog) = 5/7.
That is choice C.
7 Why not the others
A is 2/3.
Wrong: that is P(dog given girl): the other way round.
B is 2/5.
Wrong: used all 50 members: only the 28 dog lovers count.
That is the Careful mistake from before.
D is 1/2.
Wrong: girls are not half of the dog lovers.
Let's look back.
First: make a table.
Then: keep only the given column.
Then: write the fraction.
Then: simplify.
Then: check.
Then: pick the choice.
Last: why not the others.