Question

an anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. the results were as follows: 843 drank alcohol regularly, 640 smoked cigarettes, 189 used illegal drugs, 409 drank alcohol regularly and smoked cigarettes, 99 drank alcohol regularly and used illegal drugs, 101 smoked cigarettes and used illegal drugs, 81 engaged in all three behaviors, and 206 engaged in none of these behaviors. a) how many students were surveyed? of those surveyed, b) how many drank alcohol or used illegal drugs? c) how many drank alcohol only? d) how many drank alcohol and smoked cigarettes, but did not use illegal drugs? e) how many smoked cigarettes or used illegal drugs, but did not drink alcohol? f) how many engaged in exactly two of these behaviors? g) how many engaged in at least one of these behaviors?

Explanation:

Step1: Recall the principle of inclusion - exclusion

The formula for \(|A\cup B\cup C|\) is \(|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|\)

Step2: Calculate \(|A\cup B\cup C|\)

Let \(A\) be the set of students who drank alcohol, \(|A| = 843\); \(B\) be the set of students who smoked cigarettes, \(|B|=640\); \(C\) be the set of students who used illegal drugs, \(|C| = 189\); \(|A\cap B|=409\); \(|A\cap C| = 99\); \(|B\cap C|=101\); \(|A\cap B\cap C| = 81\)

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Step3: Calculate the total number of surveyed students (a)

The number of students who engaged in none of the behaviors is \(206\). So the total number of surveyed students \(N=|A\cup B\cup C|+206\)

$$N = 1144+206=1350$$

Step4: Calculate the number of students who drank alcohol or used illegal drugs (b)

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Step5: Calculate the number of students who drank alcohol only (c)

The number of students who drank alcohol only is \(|A|-(|A\cap B|+|A\cap C|)+|A\cap B\cap C|\)

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Step6: Calculate the number of students who drank alcohol and smoked cigarettes but did not use illegal drugs (d)

\(|A\cap B|-|A\cap B\cap C|=409 - 81=328\)

Step7: Calculate the number of students who smoked cigarettes or used illegal drugs but did not drink alcohol (e)

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Step8: Calculate the number of students who engaged in exactly two of these behaviors (f)

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Step9: Calculate the number of students who engaged in at least one of these behaviors (g)

\(|A\cup B\cup C|=1144\)

Answer:

a) \(1350\)
b) \(933\)
c) \(416\)
d) \(328\)
e) \(301\)
f) \(366\)
g) \(1144\)