If \( A \) and \( B \) are two events such that \( A\ne\emptyset \) , \( B\ne\emptyset \) , then which of following is/are TRUE:
A. \( P(B/A)=P(A).P(B) \)
B. \( P(B/A)=\displaystyle\frac{P(A\cap B)}{P(A)} \)
C. \( P(A'/B')=\displaystyle\frac{1-P(A\cup B)}{1-P(B)} \)
D. \( P(B/A).P(A/B)=2 \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → B and C only
By the definition of conditional probability,
\( \displaystyle P(B/A)=\frac{P(A\cap B)}{P(A)} \)
By De Morgan’s law, \( A'\cap B'=(A\cup B)' \), so
\( \displaystyle P(A'/B')=\frac{P(A'\cap B')}{P(B')} \)
\( \displaystyle =\frac{P\left((A\cup B)'\right)}{P(B')} \)
\( \displaystyle =\frac{1-P(A\cup B)}{1-P(B)} \)
So \( P(B/A)=\frac{P(A\cap B)}{P(A)} \), \( P(A'/B')=\frac{1-P(A\cup B)}{1-P(B)} \), and every conditional probability is at most 1.
Checking the statements
(A) \( P(B/A)=P(A).P(B) \) — Incorrect. \( P(B/A)=\frac{P(A\cap B)}{P(A)} \); it is \( P(A\cap B) \) that equals \( P(A)P(B) \), and only for independent events.
(B) \( P(B/A)=\frac{P(A\cap B)}{P(A)} \) — Correct. This is the definition of conditional probability.
(C) \( P(A'/B')=\frac{1-P(A\cup B)}{1-P(B)} \) — Correct. \( A'\cap B'=(A\cup B)' \) turns the definition into exactly this form.
(D) \( P(B/A).P(A/B)=2 \) — Incorrect. Each factor is at most 1, so the product cannot exceed 1.