Events A and B are such that $$P (A) = \frac{1}{2} , P(B) = \frac{7}{12}$$ and P(not A or not B) = $$\frac{1}{4}$$ . State whether A and B are independent?

$\begin{array}{1 1} \text{A and B are independent} \\\text{A and B are not independent} \end{array}$

Toolbox:
• If A and B are independent events, $$P(A\cap\;B)=P(A)\;P(B)$$
• P (not A and not B) = $$P(\bar{A}\cap\;\bar{B})$$ = $$P(\overline{A \cup B})$$ = 1 - P (A $\cup$ B)
• P (A $\cup$ B) = P(A) + P(B) - P(A $\cap$ B)
• P ($$\bar{A})$$ = 1 - P(A)
Given $$\;P(A)=\large \frac{1}{2},$$ $$\;;P(B)=\large \frac{7}{12},$$$$\;$$ and $$\;P(\bar{A}\cup\;\bar{B})=\large \frac{1}{4}$$
If A and B are independent events, $$P(A\cap\;B)=P(A)\;P(B)$$
We know that P (not A and not B) = $$P(\bar{A}\cap\;\bar{B})$$ = $$P(\overline{A \cup B})$$ = 1 - P (A $\cup$ B)
$\Rightarrow \large \frac{1}{4} $$= 1 - P (A \cup B) \rightarrow P (A \cup B) = 1 - \large \frac{1}{4} = \frac{3}{4} \Rightarrow P (A \cap B) = P (A) + P(B) - P (A \cup B) = \large \frac{1}{2} + \frac{7}{12} - \frac{3}{4} = \large \frac{6 + 7 - 4}{12} = \frac{9}{12} = \frac{3}{4} However, P(A) \times P(B) = \large \frac{1}{2} \times \large\frac{7}{12} = \frac{7}{24}$$\;\neq P (A \cap B)$
Therefore A and B are not independent.