In a bolt factory, machines A,B and C manufacture 25%, 35% and 40% respectively of the total output from which 5%, 4% and 2% respectively are defective. A bolt is drawn and is found to be defective. What is the probability that it was manufactured by the machine C?

Explanation

To solve the problem step by step, we will use Bayes' theorem, which allows us to find the probability of an event based on prior knowledge of conditions related to the event.

### Step-by-Step Solution:

1. **Identify the Probabilities:**
- Let \( P(A) \) be the probability that a bolt is manufactured by machine A.
- Let \( P(B) \) be the probability that a bolt is manufactured by machine B.
- Let \( P(C) \) be the probability that a bolt is manufactured by machine C.
- From the problem, we have:
- \( P(A) = 0.25 \)
- \( P(B) = 0.35 \)
- \( P(C) = 0.40 \)

2. **Identify the Defective Rates:**
- Let \( P(D|A) \) be the probability that a bolt is defective given it was made by machine A.
- Let \( P(D|B) \) be the probability that a bolt is defective given it was made by machine B.
- Let \( P(D|C) \) be the probability that a bolt is defective given it was made by machine C.
- From the problem, we have:
- \( P(D|A) = 0.05 \)
- \( P(D|B) = 0.04 \)
- \( P(D|C) = 0.02 \)

3. **Calculate the Total Probability of Defectiveness:**
- We need to find \( P(D) \), the total probability that a bolt is defective:
\[ P(D) = P(A) \cdot P(D|A) + P(B) \cdot P(D|B) + P(C) \cdot P(D|C) \]
- Substituting the values:
\[ P(D) = (0.25 \cdot 0.05) + (0.35 \cdot 0.04) + (0.40 \cdot 0.02) \]
- Calculating each term:
\[ P(D) = 0.0125 + 0.014 + 0.008 = 0.0345 \]

4. **Use Bayes' Theorem to Find the Probability that the Defective Bolt is from Machine C:**
- We want to find \( P(C|D) \), the probability that a defective bolt was manufactured by machine C:
\[ P(C|D) = \frac{P(D|C) \cdot P(C)}{P(D)} \]
- Substituting the known values:
\[ P(C|D) = \frac{0.02 \cdot 0.40}{0.0345} \]
- Calculating the numerator:
\[ P(C|D) = \frac{0.008}{0.0345} \]
- Finally, calculating the probability:
\[ P(C|D) \approx 0.2319 \]
- Converting to a fraction gives approximately \( \frac{16}{69} \).

5. **Final Answer:**
- The probability that a defective bolt was manufactured by machine C is \( \frac{16}{69} \).