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 information about events A and B. The question states that both A and B are non-empty events (A ≠ ∅, B ≠ ∅). We will explore the relationship between these two events using the concept of conditional probability.

### Step-by-Step Solution:

1. **Understanding Events A and B**:
- We have two events, A and B, where both events are not empty. This means that both events have some outcomes associated with them.

2. **Conditional Probability Definition**:
- The conditional probability of event A given event B is defined as:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
- Here, \(P(A | B)\) represents the probability of event A occurring given that event B has occurred.

3. **Applying the Definition**:
- Since both events A and B are non-empty, we can apply the definition of conditional probability:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
- This formula tells us how to calculate the probability of A occurring when we know that B has occurred.

4. **Interpreting the Results**:
- The intersection \(A \cap B\) represents the outcomes that are common to both events A and B.
- The probability \(P(B)\) is the probability of event B occurring.

5. **Conclusion**:
- Since both events are non-empty, \(P(B)\) is greater than 0. Therefore, we can conclude that the conditional probability \(P(A | B)\) is well-defined and can be calculated using the formula provided.

### Final Expression:
Thus, we can express the relationship between events A and B using conditional probability as follows:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]