मान लें A तथा B स्वतंत्र घटनाएँ हैं और P ( A | vedclass

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Question

It is given that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12} 	ext { and } \mathrm{P}(	ext { not } \mathrm{A} 	ext { or

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Vedclass · Answer

It is given that $\mathrm{P}(\mathrm{A})=\frac{1}{2}, \mathrm{P}(\mathrm{B})=\frac{7}{12} 	ext { and } \mathrm{P}(	ext { not } \mathrm{A} 	ext { or not } \mathrm{B})=\frac{1}{4}$.

$\Rightarrow \mathrm{P}\left(\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}ight)=\frac{1}{4}$

$\Rightarrow P\left((A \cap B)^{\prime}ight)=\frac{1}{4} \quad\left[A^{\prime} \cup B^{\prime}=(A \cap B)^{\prime}ight]$

$\Rightarrow 1-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}$

$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{3}{4}$               ........... $(1)$

However, $\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})=\frac{1}{2} \cdot \frac{7}{12}=\frac{7}{24} $          .......... $(2)$

Here, $\frac{3}{4} 
eq \frac{7}{24}$

$	herefore $ $\mathrm{P}(\mathrm{A} \cap \mathrm{B}) 
eq \mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B})$

Therefore, $A$ and $B$ are not independent events.

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