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

Let $S$ be the sample space of the given experiment

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

Given that $A$ : `head appears on the coin'
and $B{:^\prime }3$

appears on the dice'
$\Rightarrow A = \{ (H,1),(H,2),(H,3),(H,4),(H,5),(H,6)\}$ $\Rightarrow B = \{ (H,3),(T,3)\} \Rightarrow A \cap B = \{ (H,3)\}$

Hence, $P(A) = \cfrac{6}{{12}} = \cfrac{1}{2},P(B) = \cfrac{2}{{12}} = \cfrac{1}{6}$

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

$P(A) \times P(B) = \cfrac{1}{2} \times \cfrac{1}{6} = \cfrac{1}{{12}} = P(A \cap B)$

Hence, $A$ and $B$ are independent.