Suppose A and B are two events with `P(A) = 0.5 and `P(AuuB)=0.8LetP(B)=p` if A and B are mutually exclusive and P(B)=q if A and B are independent events, then value of q/p is ____.

Explanation

To solve the problem, we need to find the values of \( P \) and \( Q \) under the conditions of mutual exclusivity and independence of events \( A \) and \( B \). Then, we will calculate \( \frac{Q}{P} \).

### Step-by-Step Solution:

1. **Given Information:**
- \( P(A) = 0.5 \)
- \( P(A \cup B) = 0.8 \)

2. **Finding \( P \) when \( A \) and \( B \) are Mutually Exclusive:**
- For mutually exclusive events, \( P(A \cap B) = 0 \).
- The formula for the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the known values:
\[ 0.8 = 0.5 + P - 0 \]
- Rearranging gives:
\[ P = 0.8 - 0.5 = 0.3 \]

3. **Finding \( Q \) when \( A \) and \( B \) are Independent:**
- For independent events, \( P(A \cap B) = P(A) \cdot P(B) = 0.5 \cdot Q \).
- Using the same union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the known values:
\[ 0.8 = 0.5 + Q - (0.5 \cdot Q) \]
- Rearranging gives:
\[ 0.8 = 0.5 + Q - 0.5Q \]
- This simplifies to:
\[ 0.8 - 0.5 = Q - 0.5Q \]
\[ 0.3 = 0.5Q \]
- Solving for \( Q \):
\[ Q = \frac{0.3}{0.5} = \frac{3}{5} \]

4. **Calculating \( \frac{Q}{P} \):**
- We have \( P = 0.3 \) and \( Q = \frac{3}{5} \).
- Now, calculate \( \frac{Q}{P} \):
\[ \frac{Q}{P} = \frac{\frac{3}{5}}{0.3} \]
- Converting \( 0.3 \) to a fraction gives \( 0.3 = \frac{3}{10} \):
\[ \frac{Q}{P} = \frac{\frac{3}{5}}{\frac{3}{10}} = \frac{3}{5} \cdot \frac{10}{3} \]
- The \( 3 \) cancels out:
\[ = \frac{10}{5} = 2 \]

### Final Answer:
The value of \( \frac{Q}{P} \) is \( 2 \).