If A and B are such events that `P(A) gt 0 and P(B) ne 1 `, then P(A|B) equals

Options

A. `1 - P (A |B)`
B. `1 - P(A | B)`
C. `(1 - P (A cup B))/(P (B))`
D. `P(A ) | P(B)`

Answer

To solve the problem, we need to find \( P(A|B) \), which is the conditional probability of event A given event B. The formula for conditional probability is given by:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

where:
- \( P(A \cap B) \) is the probability that both events A and B occur.
- \( P(B) \) is the probability that event B occurs.

### Step-by-step Solution:

1. **Understanding the Given Information**:
We know that \( P(A) > 0 \) and \( P(B) \neq 1 \). This means that event A has a non-zero probability of occurring, and event B does not occur with certainty.

2. **Using the Conditional Probability Formula**:
We apply the formula for conditional probability:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

3. **Interpreting the Events**:
Since \( P(B) \neq 1 \), it implies that there is some probability that B does not occur. Therefore, we can express \( P(A \cap B) \) in terms of \( P(A) \) and \( P(B) \).

4. **Expressing \( P(A \cap B) \)**:
We can express \( P(A \cap B) \) using the law of total probability, but since we do not have specific probabilities, we keep it as \( P(A \cap B) \).

5. **Final Expression**:
Thus, we can conclude that:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

### Conclusion:
The final expression for \( P(A|B) \) is:
\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]