If `P(A)=0.3,P(B)=0.6,P(B//A)=0.5`, find `P(A uu B)`.

Answer

To find \( P(A \cup B) \) given \( P(A) = 0.3 \), \( P(B) = 0.6 \), and \( P(B|A) = 0.5 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have \( P(A) = 0.3 \)
- We have \( P(B) = 0.6 \)
- We have \( P(B|A) = 0.5 \)

2. **Use the Definition of Conditional Probability**:
- The conditional probability \( P(B|A) \) is defined as:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
- Rearranging this gives:
\[ P(A \cap B) = P(B|A) \cdot P(A) \]

3. **Calculate \( P(A \cap B) \)**:
- Substitute the values we have:
\[ P(A \cap B) = 0.5 \cdot 0.3 = 0.15 \]

4. **Use the Formula for \( P(A \cup B) \)**:
- The formula for the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

5. **Substitute the Values into the Formula**:
- Now substitute \( P(A) \), \( P(B) \), and \( P(A \cap B) \):
\[ P(A \cup B) = 0.3 + 0.6 - 0.15 \]

6. **Perform the Calculation**:
- Calculate the result:
\[ P(A \cup B) = 0.9 - 0.15 = 0.75 \]

### Final Answer:
\[ P(A \cup B) = 0.75 \]