A dice marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let $A$ be the event, the number is even and $B$ be the event, 'the number is red'. Are $A$ and $B$ independent?

The sample space is $S = \{ 1,2,3,4,5,6\}$

Give that $A:$ number is even' $\Rightarrow A = \{ 2,4,6\}$

and $B:$ 'number is red' $\Rightarrow B = \{ 1,2,3\} \Rightarrow A \cap B = \{ 2\}$

Now, $P(A) = \cfrac{3}{6} = \cfrac{1}{2},P(B) = \cfrac{3}{6} = \cfrac{1}{2}$ and $P(A \cap B) = \cfrac{1}{6}$

$\Rightarrow A$ and $B$ are not independent.