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