Question

If A and B are two events such that 2 P (A) = P (B) = \[\frac{5}{13}\]  and P (A/B) =  \[\frac{2}{5},\]  find P (A ∪ B).

Answer 1

\[\text{ Given } : \]
\[2P\left( A \right) = P\left( B \right) = \frac{5}{13} \]
\[P\left( A/B \right) = \frac{2}{5}\]
\[ \therefore P\left( A \right) = \frac{5}{26} \]
\[ P\left( B \right) = \frac{5}{13}\]
\[\text{ Now } , \]
\[ P\left( A/B \right) = \frac{P\left( A \cap B \right)}{P\left( B \right)}\]
\[ \Rightarrow \frac{2}{5} = \frac{P\left( A \cap B \right)}{\frac{5}{13}}\]
\[ \Rightarrow P\left( A \cap B \right) = \frac{2}{5} \times \frac{5}{13} = \frac{2}{13}\]
\[P\left( A \cup B \right) = P\left( A \right) + P\left( B \right) - P\left( A \cap B \right)\]
\[ = \frac{5}{26} + \frac{5}{13} - \frac{2}{13}\]
\[ = \frac{11}{26}\]