If A and B are two events such that \( \mathrm{P(A)}\ne 0,\qquad \mathrm{P(B)}\ne 1 \) and \( P\left(\bar{A}/\bar{B}\right)=\displaystyle\frac{1-k}{P(\bar{B})} \), then the value of k is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \mathrm{P(A\cup B)} \)
By the definition of conditional probability,
\( \displaystyle P(\bar{A}/\bar{B})=\frac{P(\bar{A}\cap\bar{B})}{P(\bar{B})} \)
By De Morgan’s law, \( \bar{A}\cap\bar{B}=\overline{A\cup B} \), so
\( P(\bar{A}\cap\bar{B})=1-P(A\cup B) \)
\( \displaystyle P(\bar{A}/\bar{B})=\frac{1-P(A\cup B)}{P(\bar{B})} \)
Comparing with the given form, \( k=P(A\cup B) \).