Question

If A and B be two events such that P (A) = 1/4, P (B) = 1/3 and P (A ∪ B) = 1/2, show that A and B are independent events.

Answer 1

\[P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cup B \right)\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{4} + \frac{1}{3} - \frac{1}{2}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{3 + 4 - 6}{12}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{1}{12} = \frac{1}{4} \times \frac{1}{3} = P\left( A \right)P\left( B \right)\]
\[\text{ Thus, A and B are independent events } .\]