1.A consulting firm rents cars from 3 agencies, 20% from agency A, 20% from agency B, and 60% from agency C. If 10% of the cars from A, 12% of cars from B, and 4% of the cars from C have bad tires, what is the probability that (a) the firm will get a car with bad tires? (b) a car with bad tires rented by the firm came from agency C?      

This problem can be solved using the **Total Probability Theorem** and **Bayes' Theorem**. Let's tackle each part step by step.

### Given Data:
- The firm rents cars from 3 agencies: 
  - 20% from Agency A (i.e., \( P(A) = 0.20 \))
  - 20% from Agency B (i.e., \( P(B) = 0.20 \))
  - 60% from Agency C (i.e., \( P(C) = 0.60 \))

- Probability of getting a car with bad tires:
  - From Agency A: 10% (i.e., \( P(\text{Bad} | A) = 0.10 \))
  - From Agency B: 12% (i.e., \( P(\text{Bad} | B) = 0.12 \))
  - From Agency C: 4% (i.e., \( P(\text{Bad} | C) = 0.04 \))

### (a) Probability that the firm gets a car with bad tires

The total probability of getting a car with bad tires is the weighted sum of the probabilities of getting a bad car from each agency. This can be calculated using the law of total probability:

\[
P(\text{Bad}) = P(\text{Bad} | A)P(A) + P(\text{Bad} | B)P(B) + P(\text{Bad} | C)P(C)
\]

Substitute the values:

\[
P(\text{Bad}) = (0.10)(0.20) + (0.12)(0.20) + (0.04)(0.60)
\]

Now let's compute:

\[
P(\text{Bad}) = 0.02 + 0.024 + 0.024 = 0.068
\]

Thus, the probability that the firm will get a car with bad tires is **0.068** or **6.8%**.

### (b) Probability that a car with bad tires came from Agency C

Now, we use **Bayes' Theorem** to find the probability that a car with bad tires came from Agency C. Bayes' Theorem is given by:

\[
P(C | \text{Bad}) = \frac{P(\text{Bad} | C) P(C)}{P(\text{Bad})}
\]

Substitute the known values:

\[
P(C | \text{Bad}) = \frac{(0.04)(0.60)}{0.068}
\]

Now, calculate:

\[
P(C | \text{Bad}) = \frac{0.024}{0.068} \approx 0.353
\]

Thus, the probability that a car with bad tires came from Agency C is approximately **0.353** or **35.3%**.

### Summary:
(a) The probability that the firm will get a car with bad tires is **6.8%**.

(b) The probability that a car with bad tires came from Agency C is **35.3%**.

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Would you like more details on any step?  
Here are 5 related questions for practice:

1. If Agency B had a 15% chance of bad tires, how would that affect the total probability of getting a bad car?
2. What is the probability that a car with bad tires came from Agency A?
3. How would the probabilities change if Agency C rented out 70% of the cars?
4. What is the probability that a car without bad tires came from Agency B?
5. If the firm rented 100 cars, how many would you expect to have bad tires?

**Tip:** Bayes' Theorem is powerful for updating probabilities based on new evidence—use it when given conditional information.

A consulting firm rents cars from 3 agencies, 20% from agency A, 20% from agency B, and 60% from agency C. If 10% of the cars from A, 12% of cars from B, and 4% of the cars from C have bad tires, what is the probability that (a) the firm will get a car with bad tires? (b) a car with bad tires rented by the firm came from agency C?

Explore how to calculate the probability of a consulting firm receiving a car with bad tires when renting from three different agencies. This problem applies the Total Probability Theorem and Bayes' Theorem to solve for the chances of getting a defective car, and which agency it came from. Suitable for students in Grades 10-12.

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