A fair coin and an unbiased die are tossed. Let $A$ be the event ' head appears on the coin' and $B$ be the event ' $3$ on the die'. Check whether $A$ and $B$ are independent events or not.

If a fair coin and an unbiased die are tossed, then the sample space $S$ is given by,

$S=\left\{\begin{array}{l}(H, 1),(H, 2),(H, 3),(H, 4),(H, 5),(H, 6) \\ (T, 1),(T, 2),(T, 3),(T, 4),(T, 5),(T, 6)\end{array}\right\}$

Let $A:$ Head appears on the coin

$A=\{(H, 1),(H, 2),(H, 3),(H, 4),(H, 5),(H, 6)\}$

$\Rightarrow $ $P(A)=\frac{6}{12}=\frac{1}{2}$

$\mathrm{B}: 3$ on die $=\{(\mathrm{H}, 3),(\mathrm{T}, 3)\}$

$P(B)=\frac{2}{12}=\frac{1}{6}$

$\therefore $ $A \cap B=\{(H, 3)\}$

$P(A \cap B)=\frac{1}{12}$

$P(A)\, P(B)=\frac{1}{2} \times \frac{1}{6}=P(A \cap B)$

Therefore, $A$ and $B$ are independent events.