A survey was taken of drivers regarding the factors that they considered important in buying a new car. 84 said cost. 15 said cost but not gas mileage. 72 said safety. 48 said all three. 56 said cost and safety. 25 said gas mileage but not safety. 20 said gas mileage but not cost. a. Draw the Venn diagram. b. What is the probability that someone said gas mileage in the survey? c. What is the probability that someone said both gas mileage and safety

Let's break down the problem and solve it step by step, starting with the given information:

### Variables:
Let:
- \( C \) represent those who said **cost** is important.
- \( G \) represent those who said **gas mileage** is important.
- \( S \) represent those who said **safety** is important.

The given information is:
1. \( |C| = 84 \) (84 said cost is important).
2. 15 said cost but **not** gas mileage: \( |C \cap \overline{G}| = 15 \).
3. \( |S| = 72 \) (72 said safety is important).
4. 48 said **all three**: \( |C \cap G \cap S| = 48 \).
5. 56 said cost and safety: \( |C \cap S| = 56 \).
6. 25 said gas mileage but **not** safety: \( |G \cap \overline{S}| = 25 \).
7. 20 said gas mileage but **not** cost: \( |G \cap \overline{C}| = 20 \).

### Part a: Draw the Venn Diagram

We’ll use this information to fill out a Venn diagram for the three sets \( C \), \( G \), and \( S \). To do this, we’ll work from the innermost intersection (the people who mentioned all three factors) outward.

1. \( |C \cap G \cap S| = 48 \), so there are 48 people who said **cost, gas mileage, and safety** are all important.
2. \( |C \cap S| = 56 \), but this includes the people who said all three factors are important. Therefore, the number of people who said **cost and safety but not gas mileage** is:
   \[
   |C \cap S \cap \overline{G}| = 56 - 48 = 8.
   \]
3. 15 people said **cost but not gas mileage**. This includes people who said cost and safety but not gas mileage. So the number of people who said **cost but neither safety nor gas mileage** is:
   \[
   |C \cap \overline{G} \cap \overline{S}| = 15 - 8 = 7.
   \]
4. \( |G \cap \overline{S}| = 25 \) people said **gas mileage but not safety**. This includes those who said gas mileage but neither cost nor safety, so we need to subtract those who said cost and gas mileage but not safety. The number of people who said **cost and gas mileage but not safety** is:
   \[
   |C \cap G \cap \overline{S}| = 84 - 15 = 69 \quad \text{(this includes all cost and gas mileage people)}.
   \]
   Therefore, the number of people who said **gas mileage and cost but not safety** is \( G \cap S

A survey was taken of drivers regarding the factors that they considered important in buying a new car. 84 said cost. 15 said cost but not gas mileage. 72 said safety. 48 said all three. 56 said cost and safety. 25 said gas mileage but not safety. 20 said gas mileage but not cost. a. Draw the Venn diagram. b. What is the probability that someone said gas mileage in the survey? c. What is the probability that someone said both gas mileage and safety.

Analyze a survey of drivers using Venn diagrams and probability. This problem covers set theory concepts such as intersections, complements, and the inclusion-exclusion principle. Learn how to calculate probabilities based on survey data. Suitable for high school students in Grades 9-12.

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