If `P(AcupB) = 0.8` and `P(AcapB) = 0.3`, then `P(barA)+P(barB)` is equal to

Options

A. 0.3
B. 0.9
C. 0.7
D. 0.5

Answer

To solve the problem, we need to find \( P(\bar{A}) + P(\bar{B}) \) given that \( P(A \cup B) = 0.8 \) and \( P(A \cap B) = 0.3 \).

### Step-by-Step Solution:

1. **Use the formula for the union of two events**:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We can substitute the values we know:
\[ 0.8 = P(A) + P(B) - 0.3 \]

2. **Rearrange the equation to find \( P(A) + P(B) \)**:
\[ P(A) + P(B) = 0.8 + 0.3 = 1.1 \]

3. **Express \( P(A) \) and \( P(B) \) in terms of their complements**:
\[ P(A) = 1 - P(\bar{A}) \quad \text{and} \quad P(B) = 1 - P(\bar{B}) \]
Substitute these into the equation:
\[ (1 - P(\bar{A})) + (1 - P(\bar{B})) = 1.1 \]

4. **Simplify the equation**:
\[ 2 - P(\bar{A}) - P(\bar{B}) = 1.1 \]

5. **Rearrange to find \( P(\bar{A}) + P(\bar{B}) \)**:
\[ P(\bar{A}) + P(\bar{B}) = 2 - 1.1 = 0.9 \]

### Final Answer:
\[ P(\bar{A}) + P(\bar{B}) = 0.9 \]