If `P(A) = 1/2, P(A cup B) = 3/5` and P (B) =q find the value of q is A and B are (i) Mutually exclusive (ii) independent events.

Answer

To solve the problem, we need to find the value of \( q \) (which is \( P(B) \)) under two scenarios: (i) when events A and B are mutually exclusive, and (ii) when they are independent.

### Step 1: Finding \( q \) 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:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Since \( P(A \cap B) = 0 \), the formula simplifies to:
\[ P(A \cup B) = P(A) + P(B) \]

2. **Substituting the Known Values**:
- We know:
- \( P(A) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{3}{5} \)
- Thus, substituting these values into the equation:
\[ \frac{3}{5} = \frac{1}{2} + P(B) \]

3. **Solving for \( P(B) \)**:
- Rearranging gives:
\[ P(B) = \frac{3}{5} - \frac{1}{2} \]
- To perform this subtraction, we need a common denominator (which is 10):
\[ P(B) = \frac{6}{10} - \frac{5}{10} = \frac{1}{10} \]

Thus, when A and B are mutually exclusive, \( P(B) = \frac{1}{10} \).

### Step 2: Finding \( q \) when A and B are Independent

1. **Understanding Independent Events**:
- If A and B are independent, then:
\[ P(A \cap B) = P(A) \cdot P(B) \]
- The formula for the union of two events remains:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

2. **Substituting the Values**:
- Using the independence condition, we can substitute \( P(A \cap B) \):
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]
- Plugging in the known values:
\[ \frac{3}{5} = \frac{1}{2} + P(B) - \left(\frac{1}{2} \cdot P(B)\right) \]

3. **Rearranging the Equation**:
- Let \( P(B) = q \):
\[ \frac{3}{5} = \frac{1}{2} + q - \frac{1}{2}q \]
- This simplifies to:
\[ \frac{3}{5} = \frac{1}{2} + \frac{1}{2}q \]
- Rearranging gives:
\[ \frac{3}{5} - \frac{1}{2} = \frac{1}{2}q \]

4. **Calculating the Left Side**:
- Finding a common denominator (10):
\[ \frac{3}{5} = \frac{6}{10}, \quad \frac{1}{2} = \frac{5}{10} \]
- Thus:
\[ \frac{6}{10} - \frac{5}{10} = \frac{1}{10} \]
- Therefore:
\[ \frac{1}{10} = \frac{1}{2}q \]

5. **Solving for \( q \)**:
- Multiplying both sides by 2 gives:
\[ q = \frac{2}{10} = \frac{1}{5} \]

Thus, when A and B are independent, \( P(B) = \frac{1}{5} \).

### Final Answers:
- (i) When A and B are mutually exclusive, \( P(B) = \frac{1}{10} \).
- (ii) When A and B are independent, \( P(B) = \frac{1}{5} \).