Question

If \(P(A) = \frac{3}{10}\), \(P(B) = \frac{2}{5}\) and \(P(A \bigcup B) = \frac{3}{5}\), then \(P(B|A)+P(A|B)\) is equal to:

• A. \(\frac{5}{12}\)
• B. \(\frac{7}{12}\)
• C. \(\frac{11}{12}\)
• D. \(\frac{1}{3}\)

Answer 1

The Correct Option is B

Given: \(P(A) = \frac{3}{10}\), \(P(B) = \frac{2}{5} = \frac{4}{10}\), and \(P(A \cup B) = \frac{3}{5} = \frac{6}{10}\). We need to find \(P(B|A)+P(A|B)\).

Using the formula for the union of two events: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\), we have:

\(P(A \cap B) = P(A) + P(B) - P(A \cup B) = \frac{3}{10} + \frac{4}{10} - \frac{6}{10} = \frac{1}{10}\)

The conditional probability \(P(B|A)\) is given by:

\(P(B|A) = \frac{P(A \cap B)}{P(A)} = \frac{\frac{1}{10}}{\frac{3}{10}} = \frac{1}{3}\)

The conditional probability \(P(A|B)\) is given by:

\(P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{1}{10}}{\frac{4}{10}} = \frac{1}{4}\)

Therefore, \(P(B|A) + P(A|B) = \frac{1}{3} + \frac{1}{4}\)

To add these, find a common denominator:

\(\frac{1}{3} = \frac{4}{12}\), \(\frac{1}{4} = \frac{3}{12}\)

\(\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)

Thus, \(P(B|A) + P(A|B) = \frac{7}{12}\).