Question

Mark the correct alternative in the following question:

\[\text{ If A and B are two events such that } P\left( A|B \right) = p, P\left( A \right) = p, P\left( B \right) = \frac{1}{3} \text{ and } P\left( A \cup B \right) = \frac{5}{9}, \text{ then} p = \]

Options

• \[ \frac{2}{3}\]

•  \[ \frac{3}{5}\]

• \[ \frac{1}{3} \]

• \[\frac{3}{4}\]

Answer 1

\[\text{ We have } , \]
\[P\left( A|B \right) = p, P\left( A \right) = p, P\left( B \right) = \frac{1}{3} \text{ and }  P\left( A \cup B \right) = \frac{5}{9}\]
\[\text{ As } , P\left( A|B \right) = p\]
\[ \Rightarrow \frac{P\left( A \cap B \right)}{P\left( B \right)} = p\]
\[ \Rightarrow P\left( A \cap B \right) = p \times P\left( B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = p \times \frac{1}{3}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{p}{3}\]
\[\text{ Now } , \]
\[P\left( A \cup B \right) = \frac{5}{9}\]
\[ \Rightarrow P\left( A \right) + P\left( B \right) - P\left( A \cap B \right) = \frac{5}{9}\]
\[ \Rightarrow p + \frac{1}{3} - \frac{p}{3} = \frac{5}{9}\]
\[ \Rightarrow p - \frac{p}{3} = \frac{5}{9} - \frac{1}{3}\]
\[ \Rightarrow \frac{2p}{3} = \frac{2}{9}\]
\[ \Rightarrow p = \frac{2}{9} \times \frac{3}{2}\]
\[ \therefore p = \frac{1}{3}\]