Question

Mark the correct alternative in the following question:

\[\text{ If the events A and B are independent, then }  P\left( A \cap B \right) \text{ is equal to } \]

Options

• \[P\left( A \right) + P\left( B \right)\]

• \[ P\left( A \right) - P\left( B \right) \]

• \[ P\left( A \right) P\left( B \right) \]

• \[\frac{P\left( A \right)}{P\left( B \right)}\]

Answer 1

\[\text{ As, A and B are independent events }  . \]
\[\text{ So }, P\left( A|B \right) = P\left( A \right) \text{ and}  P\left( B|A \right) = P\left( B \right)\]
\[\text{ Now } , \]
\[P\left( A|B \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)}\]
\[ \Rightarrow P\left( A \cap B \right) = P\left( B \right) P\left( A|B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = P\left( B \right) P\left( A \right)\]