In a bolt factory three machines A ,B and C manufacture `25%` , `35%` and `40%` of the total production respectively. Of their respective outputs, `5%`, `4%` and `2%` are defective. A bolt is drawn at random from the total production and it is found to be defective. Find the probability that it was manufactured by machine C.

Answer

To solve the problem step by step, we will use Bayes' theorem to find the probability that a defective bolt was manufactured by machine C.

### Step 1: Define the events
Let:
- \( E_1 \): Event that a bolt is produced by machine A
- \( E_2 \): Event that a bolt is produced by machine B
- \( E_3 \): Event that a bolt is produced by machine C
- \( D \): Event that a bolt is defective

### Step 2: Determine the probabilities of each event
From the problem statement:
- Probability of \( E_1 \) (produced by A):
\[ P(E_1) = 25\% = \frac{25}{100} = 0.25 \]
- Probability of \( E_2 \) (produced by B):
\[ P(E_2) = 35\% = \frac{35}{100} = 0.35 \]
- Probability of \( E_3 \) (produced by C):
\[ P(E_3) = 40\% = \frac{40}{100} = 0.40 \]

### Step 3: Determine the probabilities of a defective bolt given the machine
- Probability of \( D \) given \( E_1 \) (defective given produced by A):
\[ P(D|E_1) = 5\% = \frac{5}{100} = 0.05 \]
- Probability of \( D \) given \( E_2 \) (defective given produced by B):
\[ P(D|E_2) = 4\% = \frac{4}{100} = 0.04 \]
- Probability of \( D \) given \( E_3 \) (defective given produced by C):
\[ P(D|E_3) = 2\% = \frac{2}{100} = 0.02 \]

### Step 4: Use Bayes' theorem to find \( P(E_3|D) \)
Bayes' theorem states:
\[ P(E_3|D) = \frac{P(D|E_3) \cdot P(E_3)}{P(D)} \]
We need to calculate \( P(D) \) first, which is the total probability of a defective bolt:
\[ P(D) = P(D|E_1) \cdot P(E_1) + P(D|E_2) \cdot P(E_2) + P(D|E_3) \cdot P(E_3) \]
Substituting the values:
\[ P(D) = (0.05 \cdot 0.25) + (0.04 \cdot 0.35) + (0.02 \cdot 0.40) \]
Calculating each term:
- \( 0.05 \cdot 0.25 = 0.0125 \)
- \( 0.04 \cdot 0.35 = 0.014 \)
- \( 0.02 \cdot 0.40 = 0.008 \)

Adding these together:
\[ P(D) = 0.0125 + 0.014 + 0.008 = 0.0345 \]

### Step 5: Calculate \( P(E_3|D) \)
Now substitute back into Bayes' theorem:
\[ P(E_3|D) = \frac{P(D|E_3) \cdot P(E_3)}{P(D)} = \frac{0.02 \cdot 0.40}{0.0345} \]
Calculating the numerator:
\[ 0.02 \cdot 0.40 = 0.008 \]
Thus:
\[ P(E_3|D) = \frac{0.008}{0.0345} \approx 0.2319 \]

### Step 6: Simplify the fraction
To express \( P(E_3|D) \) in a simpler form:
\[ P(E_3|D) = \frac{8}{345} \]
This fraction can be simplified to:
\[ P(E_3|D) = \frac{28}{69} \]

### Final Answer
The probability that a defective bolt was manufactured by machine C is:
\[ \frac{28}{69} \]