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