Step by step
The question
Question: 60 people were asked.
32 like tea, 24 like coffee, and 10 like both.
One is picked at random.
What is the probability they like neither?
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Start in the middle
The box is all 60 people.
One circle for tea.
One circle for coffee.
10 like both.
They go in the middle, where the circles overlap.
2 Fill the rest
32 like tea, but 10 of them are already in the middle.
32 minus 10 is 22.
Coffee only: 24 minus 10 is 14.
Neither: everyone else.
60 minus 22, 10 and 14 is 14.
Careful!
Not 60 minus 32 minus 24.
That takes the 10 who like both away twice: they are in both circles.
Start in the middle.
3 Find the part we want
Neither: outside both circles, 14.
4 Write the fraction
So the probability is 14/60.
Both 14 and 60 can be divided by 2.
That makes it simpler.
So it is 7/30.
5 Check
Let's check: the four parts add up to 22 plus 10 plus 14 plus 14.
That is 60, everyone.
So the answer is right!
6 Pick the choice
Our answer is P(neither) = 7/30.
That is choice D.
7 Why not the others
A is 23/30.
Wrong: that is tea or coffee: neither is the rest.
B is 1/6.
Wrong: that is both.
C is 1/15.
Wrong: took 32 + 24 away from 60: the 10 who like both were taken away twice.
That is the Careful mistake from before.
Let's look back.
First: start in the middle.
Then: fill the rest.
Then: find the part we want.
Then: write the fraction.
Then: check.
Then: pick the choice.
Last: why not the others.