If A and B are mutually exclusive events with `P(B) ne 1, " then " P(A//overline(B))`=

Options

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

Explanation

To solve the problem, we need to find \( P(A | \overline{B}) \), where \( A \) and \( B \) are mutually exclusive events and \( P(B) \neq 1 \).

### Step-by-Step Solution:

1. **Understanding the Conditional Probability**:
We start with the formula for conditional probability:
\[ P(A | \overline{B}) = \frac{P(A \cap \overline{B})}{P(\overline{B})} \]
Here, \( P(A | \overline{B}) \) is the probability of event \( A \) occurring given that event \( B \) does not occur.

2. **Finding \( P(A \cap \overline{B}) \)**:
We can express \( P(A \cap \overline{B}) \) using the relationship between \( A \) and \( B \):
\[ P(A \cap \overline{B}) = P(A) - P(A \cap B) \]
Since \( A \) and \( B \) are mutually exclusive, \( P(A \cap B) = 0 \). Therefore:
\[ P(A \cap \overline{B}) = P(A) - 0 = P(A) \]

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

4. **Substituting into the Conditional Probability Formula**:
Now we substitute \( P(A \cap \overline{B}) \) and \( P(\overline{B}) \) back into the conditional probability formula:
\[ P(A | \overline{B}) = \frac{P(A)}{1 - P(B)} \]

5. **Final Result**:
Thus, the final expression for the conditional probability \( P(A | \overline{B}) \) is:
\[ P(A | \overline{B}) = \frac{P(A)}{1 - P(B)} \]