Question

If \(P(A)=\frac{3}{8}\), \(P(\overline{A}\mid B)=\frac{3}{5}\), \(P(\overline{B}\mid A)=\frac{3}{5}\), then \(P(A \cap B) + P(B)=\) :

• A. \( \frac{21}{40} \)
• B. \( \frac{2}{13} \)
• C. \( \frac{3}{14} \)
• D. \( \frac{5}{12} \)

Hint:

Convert all conditional probabilities into joint form first.

Answer 1

The Correct Option is A

Concept: Use conditional probability relations: \[ P(\overline{A}|B)=1-P(A|B) \]

Step 1: Find \(P(A\cap B)\).
\[ P(\overline{B}|A)=\frac{3}{5} \Rightarrow P(A|B)=\frac{2}{5} \] \[ P(A\cap B)=\frac{2}{5}P(B) \] Also: \[ P(A)=\frac{3}{8} \Rightarrow \frac{P(A\cap B)}{P(B)}=\frac{2}{5} \] \[ P(A\cap B)=\frac{3}{20} \]

Step 2: Find \(P(B)\).
\[ P(B)=\frac{5}{2}\cdot \frac{3}{20}=\frac{3}{8} \]

Step 3: Final sum.
\[ P(A\cap B)+P(B)=\frac{3}{20}+\frac{3}{8}=\frac{21}{40} \]