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

Explanation

To determine whether events A and B are independent, we need to follow these steps:

### Step 1: Understand the given probabilities
We are given:
- \( P(A) = \frac{1}{2} \)
- \( P(B) = \frac{7}{12} \)
- \( P(\text{not } A \text{ or not } B) = P(A' \cup B') = \frac{1}{4} \)

### Step 2: Use the complement rule
We know that:
\[ P(A' \cup B') = 1 - P(A \cap B) \]
Thus, we can express \( P(A \cap B) \) as:
\[ P(A \cap B) = 1 - P(A' \cup B') \]
Substituting the known value:
\[ P(A \cap B) = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 3: Calculate \( P(A) \times P(B) \)
Now, we calculate the product of the probabilities of A and B:
\[ P(A) \times P(B) = \frac{1}{2} \times \frac{7}{12} = \frac{7}{24} \]

### Step 4: Compare \( P(A \cap B) \) and \( P(A) \times P(B) \)
We have:
- \( P(A \cap B) = \frac{3}{4} \)
- \( P(A) \times P(B) = \frac{7}{24} \)

Now, we need to check if these two values are equal:
\[ \frac{3}{4} \text{ is not equal to } \frac{7}{24} \]

### Conclusion
Since \( P(A \cap B) \neq P(A) \times P(B) \), we conclude that events A and B are **not independent**.

---