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प्रश्न
Mark the correct alternative in the following question:
\[\text{ Let A and B are two events such that } P\left( A \right) = \frac{3}{8}, P\left( B \right) = \frac{5}{8} \text{ and } P\left( A \cup B \right) = \frac{3}{4} . \text{ Then } P\left( A|B \right) \times P\left( A \cap B \right) \text{ is equals to } \]
विकल्प
\[ \frac{2}{5}\]
\[ \frac{3}{8}\]
\[ \frac{3}{20}\]
\[ \frac{6}{25}\]
उत्तर
\[\text{ We have, } \]
\[P\left( A \right) = \frac{3}{8}, P\left( B \right) = \frac{5}{8} \text{ and } P\left( A \cup B \right) = \frac{3}{4}\]
\[\text{ As } , P\left( A \cup B \right) = \frac{3}{4}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = \frac{3}{4}\]
\[ \Rightarrow \frac{3}{8} + \frac{5}{8} - P\left( A \cap B \right) = \frac{3}{4}\]
\[ \Rightarrow \frac{8}{8} - P\left( A \cap B \right) = \frac{3}{4}\]
\[ \Rightarrow 1 - P\left( A \cap B \right) = \frac{3}{4}\]
\[ \Rightarrow P\left( A \cap B \right) = 1 - \frac{3}{4}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{4}\]
\[\text{ Also } , \]
\[P\left(\overline{ A } \cap B \right) = P\left( B \right) - P\left( A \cap B \right)\]
\[ = \frac{5}{8} - \frac{1}{4}\]
\[ = \frac{5 - 2}{8}\]
\[ = \frac{3}{8}\]
\[\text{ Now} , \]
\[P\left( \overline{ A }|B \right) \times P\left( A | B \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)} \times \frac{P\left( \overline{ A } \cap B \right)}{P\left( B \right)}\]
\[ = \frac{\left( \frac{1}{4} \right)}{\left( \frac{5}{8} \right)} \times \frac{\left( \frac{3}{8} \right)}{\left( \frac{5}{8} \right)}\]
\[ = \frac{8}{4 \times 5} \times \frac{3 \times 8}{5 \times 8}\]
\[ = \frac{2}{5} \times \frac{3}{5}\]
\[ = \frac{6}{25}\]
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