If A and B are two events and `A ne phi , B ne phi` then :

Options

A. `P (A//B) = P (A) . P(B)`
B. `P(A//B) = (P (A cap B))/(P(B))`
C. `P(A//B) . P (B//A) = 1`
D. `P(A//B) = P(A) // P(B)`

Answer

To solve the problem, we need to analyze the given conditions regarding the events A and B. The question states that both events A and B are not empty (A ≠ ∅ and B ≠ ∅). We are to determine the relationship between these two events and their probabilities.

### Step-by-Step Solution:

1. **Understanding the Events**:
- We are given that A and B are two events, and both are not empty. This means that there are outcomes in both events.

2. **Intersection of Events**:
- Since both events are not empty, we can consider the possibility of their intersection. The intersection of two events A and B is denoted as A ∩ B, which represents the outcomes that are common to both events.

3. **Probability of Events**:
- The probability of event A is denoted as P(A), and the probability of event B is denoted as P(B). Since both events are not empty, we have P(A) > 0 and P(B) > 0.

4. **Conditional Probability**:
- We can also consider the conditional probability of A given B, denoted as P(A|B). This is calculated using the formula:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]
- Since A and B are not empty, it is possible that P(A ∩ B) > 0, which implies that A and B may have some common outcomes.

5. **Conclusion**:
- Given that both events A and B are not empty, we can conclude that the probability of their intersection is not necessarily zero. Thus, we can say that:
- P(A ∩ B) ≥ 0
- P(A|B) is defined and can be calculated.

### Final Answer:
- The correct interpretation of the events A and B under the given conditions is that they can intersect, and their probabilities can be evaluated accordingly.