If `P(AcupB)=0.8andP(AcapB)=0.3`, then P(A') + P (B') equals to

Options

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

Answer

To solve the problem, we need to find \( P(A') + P(B') \) 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 probability of the union of two events**:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Plugging in the values we have:
\[ 0.8 = P(A) + P(B) - 0.3 \]

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

3. **Use the property of probabilities**:
We know that:
\[ P(A) + P(A') = 1 \quad \text{and} \quad P(B) + P(B') = 1 \]
Therefore, we can express \( P(A) \) and \( P(B) \) in terms of their complements:
\[ P(A) = 1 - P(A') \quad \text{and} \quad P(B) = 1 - P(B') \]

4. **Substitute these expressions into the equation**:
\[ (1 - P(A')) + (1 - P(B')) = 1.1 \]

5. **Simplify the equation**:
\[ 2 - (P(A') + P(B')) = 1.1 \]
Rearranging gives:
\[ P(A') + P(B') = 2 - 1.1 = 0.9 \]

### Final Answer:
Thus, \( P(A') + P(B') = 0.9 \).