Two companies A & B manufacture bikes. Company A manufacture 60% of bikes & B manufactures 40% of bikes, 80% bikes manufactured by company A are of super quality and other are of standard quality whereas 90% bikes manufactured by company B are of super quality and others are of standard quality. A bike is selected at random and found that it is of standard quality then find probability that bike is manufactured by company B.

Options

A. `3/4`
B. `1/5`
C. `2/3`
D. `1/4`

Answer

To solve the problem, we need to find the probability that a randomly selected standard quality bike was manufactured by company B, given that it is a standard quality bike. We can use Bayes' theorem to find this probability.

### Step-by-Step Solution:

1. **Define Events:**
- Let \( A \) be the event that the bike is manufactured by company A.
- Let \( B \) be the event that the bike is manufactured by company B.
- Let \( S \) be the event that the bike is of standard quality.

2. **Given Information:**
- Probability of selecting a bike from company A:
\[ P(A) = 0.6 \]
- Probability of selecting a bike from company B:
\[ P(B) = 0.4 \]
- Probability of a standard quality bike from company A:
\[ P(S|A) = 0.2 \quad (\text{since } 80\% \text{ are super quality, } 100\%-80\% = 20\%) \]
- Probability of a standard quality bike from company B:
\[ P(S|B) = 0.1 \quad (\text{since } 90\% \text{ are super quality, } 100\%-90\% = 10\%) \]

3. **Calculate Total Probability of Standard Quality Bikes:**
Using the law of total probability, we can find \( P(S) \):
\[ P(S) = P(A) \cdot P(S|A) + P(B) \cdot P(S|B) \]
Substituting the values:
\[ P(S) = (0.6 \cdot 0.2) + (0.4 \cdot 0.1) = 0.12 + 0.04 = 0.16 \]

4. **Apply Bayes' Theorem:**
We want to find \( P(B|S) \), the probability that the bike is manufactured by company B given that it is of standard quality:
\[ P(B|S) = \frac{P(B) \cdot P(S|B)}{P(S)} \]
Substituting the values:
\[ P(B|S) = \frac{0.4 \cdot 0.1}{0.16} \]

5. **Calculate the Result:**
\[ P(B|S) = \frac{0.04}{0.16} = \frac{1}{4} \]

### Final Answer:
The probability that a randomly selected standard quality bike was manufactured by company B is \( \frac{1}{4} \).

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