If A and B are two events such that P(A) = 0.5. P(B) = 0.6 and `P(A nn B) = 0.4`, then what is `bar(A uu B)` equal to?

Options

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

Answer

To solve the problem, we need to find the probability of the complement of the union of events A and B, denoted as \( \bar{(A \cup B)} \).

### Step-by-Step Solution:

1. **Understand the Given Values:**
- \( P(A) = 0.5 \)
- \( P(B) = 0.6 \)
- \( P(A \cap B) = 0.4 \)

2. **Use the Formula for the Probability of the Union of Two Events:**
The formula for the probability of the union of two events A and B is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

3. **Substitute the Given Values into the Formula:**
\[ P(A \cup B) = 0.5 + 0.6 - 0.4 \]

4. **Calculate \( P(A \cup B) \):**
\[ P(A \cup B) = 0.5 + 0.6 - 0.4 = 0.7 \]

5. **Find the Probability of the Complement of the Union:**
The probability of the complement of the union of A and B is given by:
\[ P(\bar{(A \cup B)}) = 1 - P(A \cup B) \]

6. **Substitute the Value of \( P(A \cup B) \):**
\[ P(\bar{(A \cup B)}) = 1 - 0.7 \]

7. **Calculate \( P(\bar{(A \cup B)}) \):**
\[ P(\bar{(A \cup B)}) = 0.3 \]

### Final Answer:
\[ P(\bar{(A \cup B)}) = 0.3 \]