Step by step
The question
Question: 40 students were asked.
22 play soccer, 15 play tennis, and 8 play both.
One is picked at random.
What is the probability they play neither?
1 Start in the middle
8 play both.
They go in the middle, where the circles overlap.
2 Fill the rest
22 play soccer, but 8 of them are already in the middle.
22 minus 8 is 14.
soccer only = 22 − 8 = 14
Tennis only: 15 minus 8 is 7.
tennis only = 15 − 8 = 7
Neither: everyone else.
40 minus 14, 8 and 7 is 11.
neither = 40 − 14 − 8 − 7 = 11
Careful!
That takes away the 8 who play both twice.
Start in the middle.
Careful: not 40 − 22 − 15 ✗
3 Find the part we want
Neither: outside both circles, 11.
4 Write the fraction
So the probability is 11/40.
P(neither) = 1140
5 Check
Let's check: the four parts add up to 14 plus 8 plus 7 plus 11.
That is 40, everyone.
So the answer is right!
14 + 8 + 7 + 11 = 40
Let's look back.
First: start in the middle.
Then: fill the rest.
Then: find the part we want.
Then: write the fraction.
Last: check.
Answer
P(neither) = 1140