If \( P(B)=\displaystyle\frac{2}{5},P(A\cup B)=\displaystyle\frac{3}{5},P(A\cap B)=\displaystyle\frac{3}{10} \) then \( P(A/B)+2\,P(B/A) \) equals
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{39}{20} \)
P(A) comes from the addition rule:
\( \displaystyle P(A)=P(A\cup B)-P(B)+P(A\cap B) \)
\( \displaystyle =\frac{3}{5}-\frac{2}{5}+\frac{3}{10} \)
\( \displaystyle =\frac{1}{2} \)
By the conditional probability formula:
\( \displaystyle P(A/B)=\frac{P(A\cap B)}{P(B)} \)
\( \displaystyle =\frac{3/10}{2/5}=\frac{3}{4} \)
\( \displaystyle P(B/A)=\frac{P(A\cap B)}{P(A)} \)
\( \displaystyle =\frac{3/10}{1/2}=\frac{3}{5} \)
\( \displaystyle P(A/B)+2\,P(B/A)=\frac{3}{4}+\frac{6}{5} \)
\( \displaystyle =\frac{15+24}{20}=\frac{39}{20} \)