KCET · Maths · Sets and Relations
If \(A\) and \(B\) are two events such that \(P(A)=\frac{1}{3}, P(B)=\frac{1}{2}\) and \(P(A \cap B)=\frac{1}{6}\), then \(P\left(A^{\prime} / B\right)\) is
- A \(\frac{2}{3}\)
- B \(\frac{1}{3}\)
- C \(\frac{1}{2}\)
- D \(\frac{1}{12}\)
Answer & Solution
Correct Answer
(A) \(\frac{2}{3}\)
Step-by-step Solution
Detailed explanation
Given \(P(A)=\frac{1}{3}, P(B)=\frac{1}{2}\) and \(P(A \cap B)=\frac{1}{6}\)
\(\begin{aligned}
P\left(A^{\prime} / B\right) &=\frac{P\left(A^{\prime} \cap B\right)}{P(B)}=\frac{P(B)-P(A \cap B)}{P(B)} \\
&=\frac{\frac{1}{2}-\frac{1}{6}}{1 / 2}=\frac{2}{3}
\end{aligned}\)
\(\begin{aligned}
P\left(A^{\prime} / B\right) &=\frac{P\left(A^{\prime} \cap B\right)}{P(B)}=\frac{P(B)-P(A \cap B)}{P(B)} \\
&=\frac{\frac{1}{2}-\frac{1}{6}}{1 / 2}=\frac{2}{3}
\end{aligned}\)
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