If A and B are events such that `P(A//B)=P(B//A)`, then

Options

A. `A sub B` but `A ne B`
B. `A=B`
C. `A nn B = phi`
D. `P(A)=P(B)`

Answer

To solve the problem, we need to derive the relationship between the probabilities of events A and B given that \( P(A|B) = P(B|A) \).

### Step-by-Step Solution:

1. **Understanding Conditional Probability**:
The conditional probability of A given B is defined as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
Similarly, the conditional probability of B given A is:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

2. **Setting Up the Equation**:
Given that \( P(A|B) = P(B|A) \), we can substitute the definitions:
\[ \frac{P(A \cap B)}{P(B)} = \frac{P(A \cap B)}{P(A)} \]

3. **Cross Multiplying**:
To eliminate the fractions, we can cross-multiply:
\[ P(A \cap B) \cdot P(A) = P(A \cap B) \cdot P(B) \]

4. **Assuming \( P(A \cap B) \neq 0 \)**:
If \( P(A \cap B) \) is not zero, we can divide both sides by \( P(A \cap B) \):
\[ P(A) = P(B) \]
This indicates that the probabilities of A and B are equal.

5. **Conclusion**:
Therefore, we conclude that if \( P(A|B) = P(B|A) \), then:
\[ P(A) = P(B) \]