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?
In this question, we are told there was a survey of 49 people. 28 of these had attended club 𝐴. This means that the probability of selecting a person that attended club 𝐴 is 28 out of 49. 38 people had attended club 𝐵. Therefore, the probability of event 𝐵 is 38 out of 49. We are also told that eight people had not been to either club. This can be written as the intersection of the complement of event 𝐴 and the complement of event 𝐵. The probability of this occurring is eight out of 49. This is represented on a Venn diagram by the area outside of our circles.
In this question, there were eight people that had not attended club 𝐴 or club 𝐵. The probability of 𝐴 union 𝐵 is equal to one minus the probability of this event, as we know that probabilities sum to one. In this question, the probability of 𝐴 union 𝐵 is equal to one minus eight over 49. This is equal to 41 over 49. There are 41 people who attended club 𝐴 or club 𝐵 or both.
We are asked to calculate the probability that a random person attended both clubs. This is the intersection of both events. We recall that the addition rule of probability states that the probability of 𝐴 union 𝐵 is equal to the probability of 𝐴 plus the probability of 𝐵 minus the probability of 𝐴 intersection 𝐵. This can be rearranged to make the probability of 𝐴 intersection 𝐵 the subject. Substituting in the values we know, this is equal to 28 over 49 plus 38 over 49 minus 41 over 49. As the denominators are the same, we simply add and subtract the numerators. The probability that a random person from the sample attended both clubs is 25 out of 49. This means that 25 people from the sample attended both clubs.
Whilst it is not required in this question, we can now complete the Venn diagram. As 28 people attended club 𝐴 and 25 of these attended both clubs, there were three people that had just attended club 𝐴. Likewise, there were 38 people who attended club 𝐵. Subtracting 25 from this gives us 13, the number of people who had just attended club 𝐵. We can check that our Venn diagram is correct by finding the sum of three, 25, 13, and eight, which gives us a total of the 49 people that were surveyed.