If \( A \) and \( B \) are any two events such that \( P(A)+P(B)-P(A\text{ and }B)=P(A) \) and \( P(B)\ne 0 \) then
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( P(A/B)=1 \)
Cancelling P(A) on both sides of the given condition:
\( \displaystyle P(B)-P(A\cap B)=0 \)
\( \displaystyle P(A\cap B)=P(B) \)
(In other words \( P(A\cup B)=P(A) \): B adds nothing outside A.) Since \( P(B)\ne 0 \), the conditional probability is
\( \displaystyle P(A/B)=\frac{P(A\cap B)}{P(B)} \)
\( \displaystyle =\frac{P(B)}{P(B)}=1 \)
Why the other options are incorrect
Option (1) \( P(B/A)=1 \) — Incorrect. \( P(B/A)=\frac{P(A\cap B)}{P(A)}=\frac{P(B)}{P(A)} \), which is 1 only when P(A) = P(B); it is not true in general.
Option (3) \( P(B/A)=0 \) — Incorrect. \( P(B/A)=\frac{P(B)}{P(A)}\ne 0 \), because \( P(B)\ne 0 \).
Option (4) \( P(A/B)=0 \) — Incorrect. \( P(A/B)=1 \), not 0.
NCERT (2026-27 edition):
Probability
