If A and B are two events such that P (A) = 0.3, P (B) = 0.6 and `P ((B)/(A))` = 0.5 find `P (A cup B)`

Answer

To solve the problem, we need to find \( P(A \cup B) \) using the provided probabilities and the formula for the union of two events.

### Step-by-Step Solution:

1. **Identify Given Values:**
- \( P(A) = 0.3 \)
- \( P(B) = 0.6 \)
- \( P(B|A) = 0.5 \)

2. **Use the Conditional Probability Formula:**
The formula for conditional probability is:
\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]
Rearranging this formula gives us:
\[ P(A \cap B) = P(B|A) \times P(A) \]

3. **Calculate \( P(A \cap B) \):**
Substitute the known values into the equation:
\[ P(A \cap B) = 0.5 \times 0.3 = 0.15 \]

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

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

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

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