If `P(A) = 1/2, P (B) = 7/12 ` and P ( not A or not B) `=1/4` , State whether A and B are independent .

Answer

To determine whether events A and B are independent, we need to check if the following condition holds:

\[ P(A \cap B) = P(A) \cdot P(B) \]

Given:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{7}{12} \)
- \( P(\text{not } A \text{ or not } B) = \frac{1}{4} \)

### Step 1: Use the complement rule
We know that:

\[ P(\text{not } A \text{ or not } B) = P(A' \cup B') = 1 - P(A \cap B) \]

### Step 2: Set up the equation
From the information given, we can set up the equation:

\[ 1 - P(A \cap B) = \frac{1}{4} \]

### Step 3: Solve for \( P(A \cap B) \)
Rearranging the equation gives us:

\[ P(A \cap B) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 4: Calculate \( P(A) \cdot P(B) \)
Now, we need to calculate \( P(A) \cdot P(B) \):

\[ P(A) \cdot P(B) = \frac{1}{2} \cdot \frac{7}{12} = \frac{7}{24} \]

### Step 5: Compare \( P(A \cap B) \) and \( P(A) \cdot P(B) \)
Now we compare the two results:

- \( P(A \cap B) = \frac{3}{4} \)
- \( P(A) \cdot P(B) = \frac{7}{24} \)

### Step 6: Determine independence
Since \( P(A \cap B) \neq P(A) \cdot P(B) \), we conclude that events A and B are not independent.

### Final Conclusion
Events A and B are **not independent**.

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