Question

Given \[ P(A)=\frac{7}{20}, \quad P(B)=\frac{1}{2}, \quad P(C)=\frac{9}{20}, \] \[ P(A\cap B\cap C)=\frac{1}{20}, \quad P(A\cap B)=\frac{3}{20}, \] \[ P(A\cap C)=\frac{1}{8}, \quad P(B\cap C)=\frac{1}{5}, \] then \(P(B|\overline{A})\) is

• A. \(\dfrac{7}{13}\)
• B. \(\dfrac{11}{20}\)
• C. \(\dfrac{19}{40}\)
• D. \(\dfrac{7}{15}\)

Hint:

Always use \[ P(B|\overline A) = \frac{P(B\cap\overline A)} {P(\overline A)} \] and \[ P(B\cap\overline A) = P(B)-P(A\cap B). \]

Answer 1

The Correct Option is A

Concept: Conditional probability is given by \[ P(B|A) = \frac{P(A\cap B)}{P(A)}. \] Similarly, \[ P(B|\overline{A}) = \frac{P(B\cap\overline{A})}{P(\overline{A})}. \]

Step 1: Find \(P(\overline A)\).
\[ P(\overline A) = 1-P(A) = 1-\frac{7}{20} = \frac{13}{20}. \]

Step 2: Find \(P(B\cap\overline A)\).
\[ P(B\cap\overline A) = P(B)-P(A\cap B). \] Substituting values, \[ = \frac12-\frac{3}{20} = \frac{10-3}{20} = \frac7{20}. \]

Step 3: Apply conditional probability formula.
\[ P(B|\overline A) = \frac{\frac7{20}} {\frac{13}{20}} = \frac7{13}. \] \[ \boxed{\frac7{13}} \]