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प्रश्न
Choose the correct alternative in the following question:
Associated to a random experiment two events A and B are such that
विकल्प
`3/10`
`1/2`
`1/10`
` 3 / 5 `
उत्तर
We have ,
\[P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and } P\left( A \cup B \right) = \frac{4}{5}\]
\[\text{ As } , P\left( A|B \right) = \frac{1}{2}\]
\[ \Rightarrow \frac{P\left( A \cap B \right)}{P\left( B \right)} = \frac{1}{2}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{2} \times P\left( B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{2} \times \frac{3}{5}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{3}{10}\]
\[\text{ Now } , \]
\[P\left( A \cup B \right) = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{3}{5} - \frac{3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{6 - 3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) = \frac{4}{5} - \frac{3}{10}\]
\[ \Rightarrow P\left( A \right) = \frac{8 - 3}{10}\]
\[ \Rightarrow P\left( A \right) = \frac{5}{10}\]
\[ \therefore P\left( A \right) = \frac{1}{2}\]
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