If `P(A//B)gt P(A)`, then which of the following is correct :

Options

A. `P(B//A)ltP(B)`
B. `P(A nn B)ltP(A).P(B)`
C. `P(B//A)gtP(B)`
D. `P(B//A)=P(B)`

Answer

To solve the problem, we start with the given condition:

**Given:** \( P(A|B) > P(A) \)

### Step 1: Write the definition of conditional probability
We know that the conditional probability \( P(A|B) \) can be expressed as:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

### Step 2: Substitute the definition into the inequality
Substituting this into our inequality gives:
\[ \frac{P(A \cap B)}{P(B)} > P(A) \]

### Step 3: Cross-multiply to eliminate the fraction
Assuming \( P(B) > 0 \), we can cross-multiply:
\[ P(A \cap B) > P(A) \cdot P(B) \]

### Step 4: Analyze the result
The inequality \( P(A \cap B) > P(A) \cdot P(B) \) suggests that the occurrence of both events A and B together is greater than what we would expect if A and B were independent.

### Step 5: Identify the implications
This implies that knowing B occurs increases the likelihood of A occurring, which is a characteristic of dependent events.

### Conclusion
From the analysis, we can conclude that:
- The correct statement among the options is that \( P(A \cap B) > P(A) \cdot P(B) \).