If event A and B are independent, such that `P(A) =3/(5), P(B) = 2/(3)," find " P(AuuB)`

Answer

To solve the problem of finding the probability of the union of two independent events A and B, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Probabilities**:
- We are given \( P(A) = \frac{3}{5} \)
- We are given \( P(B) = \frac{2}{3} \)

2. **Use the Property of Independent Events**:
- For independent events, the probability of the intersection of A and B is given by:
\[ P(A \cap B) = P(A) \times P(B) \]
- Substituting the values:
\[ P(A \cap B) = \frac{3}{5} \times \frac{2}{3} \]

3. **Calculate \( P(A \cap B) \)**:
- Performing the multiplication:
\[ P(A \cap B) = \frac{3 \times 2}{5 \times 3} = \frac{6}{15} = \frac{2}{5} \]

4. **Use the Formula for the Union of Two Events**:
- The formula for the probability of 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 substituting the known values:
\[ P(A \cup B) = \frac{3}{5} + \frac{2}{3} - \frac{2}{5} \]

6. **Find a Common Denominator**:
- The least common multiple (LCM) of the denominators (5 and 3) is 15. Convert each fraction:
\[ P(A) = \frac{3}{5} = \frac{9}{15}, \quad P(B) = \frac{2}{3} = \frac{10}{15}, \quad P(A \cap B) = \frac{2}{5} = \frac{6}{15} \]

7. **Perform the Addition and Subtraction**:
- Now substituting these values into the equation:
\[ P(A \cup B) = \frac{9}{15} + \frac{10}{15} - \frac{6}{15} \]
- Combine the fractions:
\[ P(A \cup B) = \frac{9 + 10 - 6}{15} = \frac{13}{15} \]

8. **Final Result**:
- Therefore, the probability of the union of events A and B is:
\[ P(A \cup B) = \frac{13}{15} \]