If A and B are two independent events in a sample space then `P(overline(A)//overline(B))` equals

Options

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

Answer

To find \( P(\overline{A} | \overline{B}) \) where A and B are independent events, we can follow these steps:

### Step 1: Understand the definition of conditional probability
The conditional probability of event A given event B is defined as:
\[ P(A | B) = \frac{P(A \cap B)}{P(B)} \]
In our case, we need to find \( P(\overline{A} | \overline{B}) \).

### Step 2: Use the formula for conditional probability
Using the definition of conditional probability, we can express \( P(\overline{A} | \overline{B}) \) as:
\[ P(\overline{A} | \overline{B}) = \frac{P(\overline{A} \cap \overline{B})}{P(\overline{B})} \]

### Step 3: Find \( P(\overline{A} \cap \overline{B}) \)
Since A and B are independent events, the complements \( \overline{A} \) and \( \overline{B} \) are also independent. Therefore, we can write:
\[ P(\overline{A} \cap \overline{B}) = P(\overline{A}) \cdot P(\overline{B}) \]

### Step 4: Calculate \( P(\overline{A}) \) and \( P(\overline{B}) \)
Using the property of complements:
\[ P(\overline{A}) = 1 - P(A) \]
\[ P(\overline{B}) = 1 - P(B) \]

### Step 5: Substitute back into the equation
Now substituting the values back, we have:
\[ P(\overline{A} | \overline{B}) = \frac{P(\overline{A}) \cdot P(\overline{B})}{P(\overline{B})} \]

### Step 6: Simplify the expression
Since \( P(\overline{B}) \) cancels out, we get:
\[ P(\overline{A} | \overline{B}) = P(\overline{A}) = 1 - P(A) \]

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