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मान लें $A$ तथा $B$ स्वतंत्र घटनाएँ हैं और $P ( A )=\frac{1}{2}$ तथा $P ( B )=\frac{7}{12}$ और $P ( A$ -नहीं और $B$ -नहीं $)=\frac{1}{4}$. क्या $A$ और $B$ स्वतंत्र घटनाएँ हैं?
Solution
It is given that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12} \text { and } \mathrm{P}(\text { not } \mathrm{A} \text { or not } \mathrm{B})=\frac{1}{4}$.
$\Rightarrow \mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}\right)=\frac{1}{4}$
$\Rightarrow P\left((A \cap B)^{\prime}\right)=\frac{1}{4} \quad\left[A^{\prime} \cup B^{\prime}=(A \cap B)^{\prime}\right]$
$\Rightarrow 1-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}$
$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{3}{4}$ ……….. $(1)$
However, $\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})=\frac{1}{2} \cdot \frac{7}{12}=\frac{7}{24} $ ………. $(2)$
Here, $\frac{3}{4} \neq \frac{7}{24}$
$\therefore $ $\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \neq \mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})$
Therefore, $A$ and $B$ are not independent events.
