The members of a consulting firm rent cars from three agencies: 50% from agency X, 30% from agency Y and rest from agency Z. From past experience it is known that 9% of the cars from agency X need a service and tuning before renting, 12% of the cars from agency Y need a service and tuning before renting and 10% of the cars from agency Z need a service and tuning before renting. If the rental car delivered to the firm needs service and tuning, find the probability that agency Z is not to be blamed.

Answer

To solve the problem, we will use Bayes' theorem to find the probability that agency Z is not to be blamed when a rental car needs service and tuning. Let's break down the solution step by step.

### Step 1: Define Events
Let:
- \( A \): the event that a car needs service.
- \( E_1 \): the event that the car is rented from agency X.
- \( E_2 \): the event that the car is rented from agency Y.
- \( E_3 \): the event that the car is rented from agency Z.

### Step 2: Assign Probabilities
From the problem statement, we have:
- \( P(E_1) = 0.50 \) (50% from agency X)
- \( P(E_2) = 0.30 \) (30% from agency Y)
- \( P(E_3) = 0.20 \) (20% from agency Z, since it is the remainder)

The probabilities that a car from each agency needs service are:
- \( P(A | E_1) = 0.09 \) (9% from agency X)
- \( P(A | E_2) = 0.12 \) (12% from agency Y)
- \( P(A | E_3) = 0.10 \) (10% from agency Z)

### Step 3: Calculate Total Probability of A
Using the law of total probability, we calculate \( P(A) \):
\[ P(A) = P(A | E_1) P(E_1) + P(A | E_2) P(E_2) + P(A | E_3) P(E_3) \]
Substituting the values:
\[ P(A) = (0.09 \times 0.50) + (0.12 \times 0.30) + (0.10 \times 0.20) \]
Calculating each term:
\[ P(A) = 0.045 + 0.036 + 0.020 = 0.101 \]

### Step 4: Calculate \( P(E_3 | A) \)
Now, we need to find \( P(E_3 | A) \) using Bayes' theorem:
\[ P(E_3 | A) = \frac{P(A | E_3) P(E_3)}{P(A)} \]
Substituting the values:
\[ P(E_3 | A) = \frac{(0.10 \times 0.20)}{0.101} = \frac{0.02}{0.101} \]
Calculating this gives:
\[ P(E_3 | A) = \frac{20}{101} \]

### Step 5: Find Probability that Agency Z is Not to be Blamed
We want the probability that agency Z is not to be blamed when a car needs service, which is \( P(E_3^c | A) \):
\[ P(E_3^c | A) = 1 - P(E_3 | A) \]
Substituting the value we found:
\[ P(E_3^c | A) = 1 - \frac{20}{101} = \frac{101 - 20}{101} = \frac{81}{101} \]

### Final Answer
Thus, the probability that agency Z is not to be blamed when a rental car needs service is:
\[ \frac{81}{101} \]