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 of finding \( P(\overline{A} | B) \), we can use the definition of conditional probability. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Understanding Conditional Probability**:
The conditional probability of event \( A \) given event \( B \) is defined as:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
Similarly, for \( \overline{A} \) (the complement of \( A \)), we have:
\[ P(\overline{A} | B) = \frac{P(\overline{A} \cap B)}{P(B)} \]

2. **Using the Complement Rule**:
We know that:
\[ P(A \cap B) + P(\overline{A} \cap B) = P(B) \]
This means we can express \( P(\overline{A} \cap B) \) as:
\[ P(\overline{A} \cap B) = P(B) - P(A \cap B) \]

3. **Expressing \( P(A \cap B) \)**:
From the definition of conditional probability, we can express \( P(A \cap B) \) as:
\[ P(A \cap B) = P(A | B) \cdot P(B) \]

4. **Substituting Back**:
Now, substituting \( P(A \cap B) \) back into the 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. **Finding \( P(\overline{A} | B) \)**:
Now, substituting \( P(\overline{A} \cap B) \) into the formula for \( P(\overline{A} | B) \):
\[ P(\overline{A} | B) = \frac{P(\overline{A} \cap B)}{P(B)} = \frac{P(B)(1 - P(A | B))}{P(B)} = 1 - P(A | B) \]

6. **Final Result**:
Therefore, we conclude that:
\[ P(\overline{A} | B) = 1 - P(A | B) \]