Given that the events A and B are such that `P(A)=1/2, P(AuuB)=3/5a n d\ P(B)=pdot` Find `p` if they are i. mutually exclusive, ii. independent.

Explanation

To solve the problem, we need to find the value of \( p \) given the probabilities of events A and B under two different conditions: mutually exclusive and independent.

### Given:
- \( P(A) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{3}{5} \)
- \( P(B) = p \)

### Part 1: When A and B are Mutually Exclusive

1. **Understanding Mutually Exclusive Events**:
- If A and B are mutually exclusive, then \( P(A \cap B) = 0 \).
- The formula for the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting \( P(A \cap B) = 0 \):
\[ P(A \cup B) = P(A) + P(B) \]

2. **Substituting the Given Values**:
- We know \( P(A \cup B) = \frac{3}{5} \), \( P(A) = \frac{1}{2} \), and \( P(B) = p \):
\[ \frac{3}{5} = \frac{1}{2} + p \]

3. **Solving for p**:
- Rearranging the equation:
\[ p = \frac{3}{5} - \frac{1}{2} \]
- Finding a common denominator (which is 10):
\[ p = \frac{6}{10} - \frac{5}{10} = \frac{1}{10} \]

### Part 2: When A and B are Independent

1. **Understanding Independent Events**:
- If A and B are independent, then:
\[ P(A \cap B) = P(A) \cdot P(B) \]

2. **Using the Union Formula**:
- We can still use the union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting for \( P(A \cap B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

3. **Substituting the Given Values**:
- Plugging in the known values:
\[ \frac{3}{5} = \frac{1}{2} + p - \left(\frac{1}{2} \cdot p\right) \]

4. **Rearranging the Equation**:
- Simplifying the equation:
\[ \frac{3}{5} = \frac{1}{2} + p - \frac{p}{2} \]
- Combine like terms:
\[ \frac{3}{5} = \frac{1}{2} + \frac{2p}{2} - \frac{p}{2} = \frac{1}{2} + \frac{p}{2} \]

5. **Finding a Common Denominator**:
- The common denominator is 10:
\[ \frac{3}{5} = \frac{6}{10} \quad \text{and} \quad \frac{1}{2} = \frac{5}{10} \]
- Thus:
\[ \frac{6}{10} = \frac{5}{10} + \frac{p}{2} \]

6. **Solving for p**:
- Rearranging gives:
\[ \frac{p}{2} = \frac{6}{10} - \frac{5}{10} = \frac{1}{10} \]
- Therefore:
\[ p = \frac{1}{10} \cdot 2 = \frac{1}{5} \]

### Final Answers:
- For mutually exclusive events, \( p = \frac{1}{10} \).
- For independent events, \( p = \frac{1}{5} \).