A survey asked 49 people if they had visited any clubs recently. 28 had attended Club A, 38 had attended Club B, and eight had not been to either club. What is the probability that a random person from the sample attended both clubs?
There are lots of ways to approach this problem, but we are going to look at it through a Venn diagram. The left-hand circle will include all the people that attended Club A. The right-hand circle will include all the people that attended Club B. And the intersection of the two circles will be the people that attended both.
As the Venn diagram will contain all 49 people, outside of the circles will be the eight people that haven’t been to either club. There were 28 people that attended Club A. So we can write the number 28 inside the left-hand circle. 38 people had attended Club B. This means we can write 38 inside the right-hand circle. As eight people had not attended either club, we can write the eight outside of the circles.
You’ll notice at present that there is no number inside the intersection of circle A and circle B. Also when we had the three numbers 28, 38, and eight, we get an answer of 74. However, there were only 49 people in the survey. This means that we need to subtract 49 from 74 as some of the people have been counted twice — the people that attended Club A and Club B. 74 minus 49 is equal to 25. This means that there were 25 people that attended both Club A and Club B.
Whilst we could answer the question “what is the probability that a random person attended both clubs?” from the diagram, it is important that we complete the Venn diagram accurately. If we consider circle A, we know that 25 people attended Club A and Club B. There were 28 people in total that attended Club A. Therefore, the number of people that attended only Club A was three as three plus 25 is equal to 28. We can do the same thing for the 38 people that attended Club B: 38 minus 25 is equal to 13. Therefore, 13 people attended Club B only.
We can check that our diagram is correct by adding the four numbers: 25 plus three plus 13 plus eight. As this is equal to 49, we know that our Venn diagram is correct.
The probability of an event occurring is the number of successful outcomes divided by the number of possible outcomes. In this case, the probability that a person attended both clubs is 25 out of 49 or twenty-five forty-ninths as they were 25 people that attended both clubs and 49 people who took part in the survey in total.