14.ProbabilityMedium

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'. Are $A$ and $B$ independent?

Solution

When a die is thrown, the sample space ( $S$ ) is

$\mathrm{S}=\{1,2,3,4,5,6\}$

Let $A:$ the number is even $=\{2,4,6\}$

$\Rightarrow P(A)=\frac{3}{6}=\frac{1}{2}$

$B:$ the number is red $=\{1,2,3\}$

$\Rightarrow P(B)=\frac{3}{6}=\frac{1}{2}$

$\therefore $ $A \cap B=\{2\}$

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

$P(A) P(B)=\frac{1}{2} \times \frac{1}{2}=\frac{1}{4} \neq \frac{1}{6}$

$\Rightarrow $ $P(A) \cdot P(B) \neq P(A B)$

Therefore, $A$ bad $B$ are not independent.

Events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12}$ and $\mathrm{P}$ $($ not $ \mathrm{A}$ or not $\mathrm{B})=\frac{1}{4} .$ State whether $\mathrm{A}$ and $\mathrm{B}$ are independent?

Medium

Medium

Difficult

Easy

Medium