Suppose A and B are two events. Event B has occurred and it is known that `P(B) lt 1`. What is `P(A//B')` equal to?

Options

A. `(P(A)-P(B))/(1-P(B))`
B. `(P(A)-P(AB))/(1-P(B))`
C. `(P(A)-P(B'))/(1-P(B))`
D. None of these

Answer

To solve the problem, we need to find the conditional probability \( P(A | B') \), where \( B' \) is the complement of event \( B \).

### Step-by-Step Solution:

1. **Understanding the Events**:
- We have two events A and B.
- We know that event B has occurred and \( P(B) < 1 \). This means there is a non-zero probability for the complement of B, denoted as \( B' \).

2. **Using the Definition of Conditional Probability**:
- The conditional probability \( P(A | B') \) is defined as:
\[ P(A | B') = \frac{P(A \cap B')}{P(B')} \]

3. **Finding \( P(B') \)**:
- The probability of the complement of B is given by:
\[ P(B') = 1 - P(B) \]

4. **Finding \( P(A \cap B') \)**:
- We can express \( P(A \cap B') \) in terms of \( P(A) \) and \( P(A \cap B) \):
\[ P(A \cap B') = P(A) - P(A \cap B) \]

5. **Substituting into the Conditional Probability Formula**:
- Now we can substitute \( P(A \cap B') \) and \( P(B') \) into the formula for \( P(A | B') \):
\[ P(A | B') = \frac{P(A) - P(A \cap B)}{1 - P(B)} \]

6. **Final Expression**:
- Thus, the final expression for \( P(A | B') \) is:
\[ P(A | B') = \frac{P(A) - P(A \cap B)}{1 - P(B)} \]

### Conclusion:
The conditional probability \( P(A | B') \) can be expressed as:
\[ P(A | B') = \frac{P(A) - P(A \cap B)}{1 - P(B)} \]