In a bolt factory, machines `A,B` and `C` manufacture respectively `20%,30%` and `50%` of the total bolt. Of their output `3,4` and `2` percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine `C` is

Options

A. `(3)/(7)`
B. `(9)/(28)`
C. `(5)/(14)`
D. `(2)/(7)`

Answer

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

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

### Step 2: Determine the probabilities of each machine producing a bolt
From the problem, we know:
- \( P(A) = \frac{20}{100} = 0.2 \)
- \( P(B) = \frac{30}{100} = 0.3 \)
- \( P(C) = \frac{50}{100} = 0.5 \)

### Step 3: Determine the probabilities of a bolt being defective given the machine
We also know the percentages of defective bolts from each machine:
- \( P(D|A) = \frac{3}{100} = 0.03 \)
- \( P(D|B) = \frac{4}{100} = 0.04 \)
- \( P(D|C) = \frac{2}{100} = 0.02 \)

### Step 4: Use the law of total probability to find \( P(D) \)
We need to calculate the total probability of drawing a defective bolt:
\[ P(D) = P(D|A)P(A) + P(D|B)P(B) + P(D|C)P(C) \]
Substituting the values:
\[ P(D) = (0.03)(0.2) + (0.04)(0.3) + (0.02)(0.5) \]
Calculating each term:
\[ P(D) = 0.006 + 0.012 + 0.01 = 0.028 \]

### Step 5: Apply Bayes' theorem to find \( P(C|D) \)
We want to find the probability that a defective bolt was produced by machine C:
\[ P(C|D) = \frac{P(D|C)P(C)}{P(D)} \]
Substituting the known values:
\[ P(C|D) = \frac{(0.02)(0.5)}{0.028} \]
Calculating the numerator:
\[ P(C|D) = \frac{0.01}{0.028} \]
Now, simplifying:
\[ P(C|D) = \frac{100}{280} = \frac{5}{14} \]

### Final Answer
Thus, the probability that a defective bolt is manufactured by machine C is \( \frac{5}{14} \).

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