An automobile manufacturer is concerned about a possible recall of its best-selling 
four-door sedan. If there were a recall, there is a probability of 0.25 of a defect in 
the brake system, 0.18 of a defect in the transmission, 0.17 of a defect in the fuel 
system, and 0.40 of a defect in some other area.
(a) What is the probability that the defect is the brakes or the fuelling system if the 
probability of defects in both systems simultaneously is 0.15?

To solve the problem, we need to determine the probability that the defect is in the brake system **or** the fuel system. This can be calculated using the **principle of inclusion-exclusion**.

### Given:
- Probability of a defect in the brake system, \( P(B) = 0.25 \)
- Probability of a defect in the fuel system, \( P(F) = 0.17 \)
- Probability of defects in both systems simultaneously, \( P(B \cap F) = 0.15 \)

### Formula:
The probability that the defect is in either the brake system or the fuel system is given by:
\[
P(B \cup F) = P(B) + P(F) - P(B \cap F)
\]

### Substituting the values:
\[
P(B \cup F) = 0.25 + 0.17 - 0.15
\]

Let's calculate this.The probability that the defect is in either the brake system or the fuel system is **0.27**.

Would you like more details or have any questions? Here are 5 questions that could expand this analysis:

1. How would the probability change if the defect probabilities were dependent on each other?
2. What is the probability that the defect is in neither the brake system nor the fuel system?
3. How would you compute the probability of a defect in either the transmission or the fuel system?
4. What if you had more systems to consider? How would you extend this approach to multiple events?
5. How does understanding these probabilities help in making decisions about recalls?

**Tip:** In problems involving multiple events, the principle of inclusion-exclusion is a powerful tool to avoid over-counting when calculating the probability of the union of events.

Learn how to calculate the probability of defects in either the brake system or the fuel system using the inclusion-exclusion principle. This problem involves understanding probabilities in automotive recalls and applies mathematical concepts like probability and set theory. Suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation