Question

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.

• A. True
• B. False

Answer 1

The correct option is A True

If a fair coin and an unbiased die are tossed, then the sample space $S$ is given by,
$S=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6),(T,1),(T,2),(T,3),(T,4),(T,5),(T,6)\}$
Let $A$ : Head appears on the coin
$A=\{(H,1),(H,2),(H,3),(H,4),(H,5),(H,6)\}$

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

$B$ : $3$ on die $=\{(H,3),(T,3)\}$
$P\left(B\right)=\frac{2}{12}=\frac{1}{6}$
$\therefore A\cap B=\left\{(H,3)\right\}$

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

$P\left(A\right)\cdot P\left(B\right)=\frac{1}{2}\times \frac{1}{6}=P(A\cap B)$
Therefore, $A$ and $B$ are independent events.