Question

State true or false.

A die 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'. Then $A$ and $B$ are independent events.

• A. True
• B. False

Answer 1

The correct option is B False

When a die is thrown, the sample space $\left(S\right)$ is $S=\{1,2,3,4,5,6\}$
Let $A$ : the number is even $=\{2,4,6\}$
$\Rightarrow P\left(A\right)=\frac{3}{6}=\frac{1}{2}$
$B$ : the number is red $=\{1,2,3\}$
$\Rightarrow P\left(B\right)=\frac{3}{6}=\frac{1}{2}$
$\therefore A\cap B=\left\{2\right\}$
$P\left(AB\right)=P(A\cap B)=\frac{1}{6}$
$P\left(A\right)\cdot P\left(B\right)=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}\ne \frac{1}{6}$
$\Rightarrow P\left(A\right)\cdot P\left(B\right)\ne P\left(AB\right)$
Therefore, $A$ and $B$ are not independent.