If A and B are two events such that \(P(B)=\displaystyle\frac{3}{5}\), \(P(A|B)=\displaystyle\frac{1}{2}\) and \(P(A \cup B)=\displaystyle\frac{4}{5}\), then P(A) is equal to :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{1}{2}\)
By the definition of conditional probability, \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), so
\( \displaystyle P(A \cap B) = P(A|B)\times P(B) \)
\( \displaystyle = \frac{1}{2}\times\frac{3}{5} = \frac{3}{10} \)
By the addition rule:
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
\( \displaystyle \frac{4}{5} = P(A) + \frac{3}{5} - \frac{3}{10} \)
\( \displaystyle P(A) = \frac{4}{5} - \frac{3}{5} + \frac{3}{10} \)
\( \displaystyle = \frac{2}{10} + \frac{3}{10} = \frac{1}{2} \)