Events `Aa n dC` are independent. If the probabilities relating `A ,B ,a n dC` are `P(A)=1//5, P(B)=1//6; P(AnnC)=1//20 ; P(BuuC)=3//8.` Then events `a)Ba n dC` are independent events `b)Ba n dC` are mutually exclusive events `c)Ba n dC` are neither independent nor mutually exclusive events `d)Ba n dC` are equiprobable

Options

A. events B and C are independent
B. events B and C are mutually exclusive
C. events B and C are neiter independent nor mutually exclusive
D. events B and C are equiprobabile

Explanation

To solve the problem, we need to analyze the given probabilities and determine the relationship between events B and C.

### Step-by-Step Solution:

1. **Given Probabilities**:
- \( P(A) = \frac{1}{5} \)
- \( P(B) = \frac{1}{6} \)
- \( P(A \cap C) = \frac{1}{20} \)
- \( P(B \cup C) = \frac{3}{8} \)

2. **Finding \( P(C) \)**:
Since A and C are independent, we can use the formula for independent events:
\[ P(A \cap C) = P(A) \cdot P(C) \]
Substituting the known values:
\[ \frac{1}{20} = \frac{1}{5} \cdot P(C) \]
To find \( P(C) \):
\[ P(C) = \frac{1}{20} \div \frac{1}{5} = \frac{1}{20} \cdot \frac{5}{1} = \frac{1}{4} \]

3. **Finding \( P(B \cap C) \)**:
We can use the formula for the union of two events:
\[ P(B \cup C) = P(B) + P(C) - P(B \cap C) \]
Substituting the known values:
\[ \frac{3}{8} = \frac{1}{6} + \frac{1}{4} - P(B \cap C) \]
First, we need a common denominator to add \( \frac{1}{6} \) and \( \frac{1}{4} \). The least common multiple of 6 and 4 is 12.
- Convert \( \frac{1}{6} \) to twelfths: \( \frac{1}{6} = \frac{2}{12} \)
- Convert \( \frac{1}{4} \) to twelfths: \( \frac{1}{4} = \frac{3}{12} \)

Now substituting:
\[ \frac{3}{8} = \frac{2}{12} + \frac{3}{12} - P(B \cap C) \]
Combine the fractions:
\[ \frac{3}{8} = \frac{5}{12} - P(B \cap C) \]
Rearranging gives:
\[ P(B \cap C) = \frac{5}{12} - \frac{3}{8} \]
To subtract these fractions, we need a common denominator. The least common multiple of 12 and 8 is 24.
- Convert \( \frac{5}{12} \) to twenty-fourths: \( \frac{5}{12} = \frac{10}{24} \)
- Convert \( \frac{3}{8} \) to twenty-fourths: \( \frac{3}{8} = \frac{9}{24} \)

Now substituting:
\[ P(B \cap C) = \frac{10}{24} - \frac{9}{24} = \frac{1}{24} \]

4. **Checking Independence of B and C**:
We need to check if \( P(B \cap C) = P(B) \cdot P(C) \):
\[ P(B) \cdot P(C) = \frac{1}{6} \cdot \frac{1}{4} = \frac{1}{24} \]
Since \( P(B \cap C) = \frac{1}{24} \) and \( P(B) \cdot P(C) = \frac{1}{24} \), we conclude that B and C are independent.

5. **Conclusion**:
Since we have established that B and C are independent, we can select the correct answer from the options given:
- a) B and C are independent (Correct)
- b) B and C are mutually exclusive (Incorrect)
- c) B and C are neither independent nor mutually exclusive (Incorrect)
- d) B and C are equiprobable (Incorrect)

### Final Answer:
**Option a) B and C are independent.**