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Two events $A$ and $B$ will be independent, if
$A$ and $B$ are mutually exclusive
$P\left(A^{\prime} B^{\prime}\right)=[1-P(A)][1-P(B)]$
$P(A)=P(B)$
$P(A)+P(B)=1$
Solution
Two events $A$ and $B$ are said to be independent, if $P(A B)=P(A) \times P(B)$
Consider the result given in alternative $B$.
$\mathrm{P}\left(\mathrm{A} \mathrm{B}^{\prime}\right)=[1-\mathrm{P}(\mathrm{A})][1-\mathrm{P}(\mathrm{B})]$
$\Rightarrow \mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{A}) . \mathrm{P}(\mathrm{B})$
$\Rightarrow 1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})=1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{AB})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{AB})=\mathrm{P}(\mathrm{A}). \mathrm{P}(\mathrm{B})$
This implies that $\mathrm{A}$ and $\mathrm{B}$ are independent, if $\mathrm{P}\left(\mathrm{AB}^{\prime}\right)=[1-\mathrm{P}(\mathrm{A})][1-\mathrm{P}(\mathrm{B})]$
Distracter Rationale
$A.$ Let $\mathrm{P}(\mathrm{A})=\mathrm{m}, \,\mathrm{P}(\mathrm{B})=\mathrm{n}, \,0<\mathrm{m}, \,\mathrm{n}<1$
$A$ and $B$ are mutually exclusive.
$\therefore \mathrm{A} \cap \mathrm{B}=\phi$
$\Rightarrow \mathrm{P}(\mathrm{AB})=0$
However, $\mathrm{P}(\mathrm{A}) . \mathrm{P}(\mathrm{B})=\mathrm{mn} \neq 0$
$\therefore \mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B}) \neq \mathrm{P}(\mathrm{AB})$
$C.$ Let $A:$ Event of getting an odd number on throw of a die $=\{1,3,5\}$
$\Rightarrow P(A)=\frac{3}{6}=\frac{1}{2}$
$B:$ Event of getting an even number on throw of a die $=\{2,4,6\}$
$P(B)=\frac{3}{6}=\frac{1}{2}$
Here, $A \cap B=\phi$
$\mathrm{P}(\mathrm{AB})=0$
$P(A) \cdot P(B)=\frac{1}{4} \neq 0$
$\mathrm{P}(\mathrm{A}) .\mathrm{P}(\mathrm{B}) \neq \mathrm{P}(\mathrm{AB})$
$D.$ From the above example, it can be seen that, $P(A)+P(B)=\frac{1}{2}+\frac{1}{2}=1$
However, it cannot be inferred that $\mathrm{A}$ and $\mathrm{B}$ are independent.
Thus, the correct answer is $B$.