If A and B are two events such that `P(A)gt0 and P(B)!=1,` then `P((overline(A))/(B))` is equal to

Answer

To solve the problem, we need to find \( P(\overline{A} | B) \), which is the probability of the complement of event A given event B.

### Step-by-Step Solution:

1. **Understanding the Conditional Probability**:
The conditional probability \( P(\overline{A} | B) \) can be expressed using the formula:
\[ P(\overline{A} | B) = \frac{P(\overline{A} \cap B)}{P(B)} \]

2. **Using the Complement Rule**:
We know that the probability of an event and its complement sum up to 1. Therefore, we can express \( P(A) \) in terms of \( P(\overline{A}) \):
\[ P(A) + P(\overline{A}) = 1 \implies P(\overline{A}) = 1 - P(A) \]

3. **Finding \( P(\overline{A} \cap B) \)**:
To find \( P(\overline{A} \cap B) \), we can use the law of total probability:
\[ P(\overline{A} \cap B) = P(B) - P(A \cap B) \]
However, we can also express \( P(A \cap B) \) in terms of conditional probability:
\[ P(A \cap B) = P(A | B) \cdot P(B) \]

4. **Substituting Back**:
Now, substituting \( P(A \cap B) \) back into our equation for \( P(\overline{A} \cap B) \):
\[ P(\overline{A} \cap B) = P(B) - P(A | B) \cdot P(B) \]
This simplifies to:
\[ P(\overline{A} \cap B) = P(B)(1 - P(A | B)) \]

5. **Final Calculation**:
Now we can substitute this into our conditional probability formula:
\[ P(\overline{A} | B) = \frac{P(\overline{A} \cap B)}{P(B)} = \frac{P(B)(1 - P(A | B))}{P(B)} \]
Since \( P(B) \neq 0 \), we can cancel \( P(B) \):
\[ P(\overline{A} | B) = 1 - P(A | B) \]

### Conclusion:
Thus, the final result is:
\[ P(\overline{A} | B) = 1 - P(A | B) \]