If \(P(A) = \displaystyle\frac{3}{10}, P(B) = \displaystyle\frac{2}{5}\) and \(P(A\cup B) = \displaystyle\frac{3}{5}\), then the value of \(P(B|A) + P(A|B)\) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{7}{12}\)
By the addition theorem of probability:
\(P(A\cap B) = P(A) + P(B) - P(A\cup B)\)
\(\displaystyle = \frac{3}{10} + \frac{4}{10} - \frac{6}{10} = \frac{1}{10}\)
By the definition of conditional probability:
\(\displaystyle P(B|A) = \frac{P(A\cap B)}{P(A)}\)
\(\displaystyle = \frac{1/10}{3/10} = \frac{1}{3}\)
\(\displaystyle P(A|B) = \frac{P(A\cap B)}{P(B)}\)
\(\displaystyle = \frac{1/10}{4/10} = \frac{1}{4}\)
\(\displaystyle P(B|A) + P(A|B) = \frac{1}{3} + \frac{1}{4} = \frac{7}{12}\)