Question

Choose the correct alternative in the following question:
Associated to a random experiment two events A and B are such that

\[P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ an d }  P\left( A \cup B \right) = \frac{4}{5}\] . The value of P(A) is

विकल्प

• `3/10`

• `1/2`

• `1/10`

• ` 3 / 5 `

Answer 1

We have  ,
\[P\left( B \right) = \frac{3}{5}, P\left( A|B \right) = \frac{1}{2} \text{ and }  P\left( A \cup B \right) = \frac{4}{5}\]
\[\text{ As } , P\left( A|B \right) = \frac{1}{2}\]
\[ \Rightarrow \frac{P\left( A \cap B \right)}{P\left( B \right)} = \frac{1}{2}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{2} \times P\left( B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{2} \times \frac{3}{5}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{3}{10}\]
\[\text{ Now } , \]
\[P\left( A \cup B \right) = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{3}{5} - \frac{3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{6 - 3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) + \frac{3}{10} = \frac{4}{5}\]
\[ \Rightarrow P\left( A \right) = \frac{4}{5} - \frac{3}{10}\]
\[ \Rightarrow P\left( A \right) = \frac{8 - 3}{10}\]
\[ \Rightarrow P\left( A \right) = \frac{5}{10}\]
\[ \therefore P\left( A \right) = \frac{1}{2}\]