Question

A die marked $1,2,3$ in red and $4,5,6$ in green is tossed. Let $A$ be the event "number is even" and $B$ be the event "number is marked red". Find whether the events $A$ and $B$ are independent or not.

Answer 1

According to the question,
$A=\{2,4,6\}\&B=\{1,2,3\}$
$\Rightarrow A\cap B=\left\{2\right\}P\left(A\right)=\frac{3}{6}=\frac{1}{2},P\left(B\right)=\frac{3}{6}=\frac{1}{2},P(A\cap B)=\frac{1}{6}P\left(A\right).P\left(B\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}\ne P(A\cap B)$
$\therefore A\&B$ are not independent events.