Math / Data & Statistics

QuestionAn anonymous survey of college students was taken to determine behaviors regarding alcohol, cigarettes, and illegal drugs. The results were as follows: 835 drank alcohol regularly, 655 smoked cigarettes, 205 used illegal drugs, 401 drank alcohol regularly and smoked cigarettes, 113 drank alcohol regularly and used illegal drugs, 126 smoked cigarettes and used illegal drugs, 96 engaged in all three behaviors, and 349 engaged in none of these behaviors. a) How many students were surveyed?

Studdy Solution

STEP 1

1. The survey results can be represented using set theory.
2. The principle of inclusion-exclusion will be used to calculate the total number of students surveyed.

STEP 2

1. Define sets for each behavior.
2. Apply the principle of inclusion-exclusion to find the total number of students.

STEP 3

Define the sets: - Let $A$ be the set of students who drank alcohol regularly. - Let $B$ be the set of students who smoked cigarettes. - Let $C$ be the set of students who used illegal drugs.
Given: $|A| = 835, \quad |B| = 655, \quad |C| = 205$ $|A \cap B| = 401, \quad |A \cap C| = 113, \quad |B \cap C| = 126$ $|A \cap B \cap C| = 96$ $|A' \cap B' \cap C'| = 349$

STEP 4

Apply the principle of inclusion-exclusion to find the total number of students surveyed:
The formula for the union of three sets is: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$
Substitute the given values: $|A \cup B \cup C| = 835 + 655 + 205 - 401 - 113 - 126 + 96$
Calculate: $|A \cup B \cup C| = 1695 - 640 + 96 = 1151$

STEP 5

Add the students who engaged in none of these behaviors to find the total surveyed: $\text{Total surveyed} = |A \cup B \cup C| + |A' \cap B' \cap C'|$ $\text{Total surveyed} = 1151 + 349 = 1500$
The total number of students surveyed is:
$\boxed{1500}$

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