If A and B are any two events such that P(B) = P(A and B), then which of the following is correct
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(P(A|B) = 1\)
Here P(A and B) means \(P(A \cap B)\). By the definition of conditional probability (with P(B) ≠ 0, as needed for P(A|B) to be defined):
\( \displaystyle P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(B)}{P(B)} = 1 \)
In words: whenever B occurs, A also occurs.
Why the other options are incorrect
Option (1) \(P(B|A) = 1\) — Incorrect. This reverses the condition: \(P(B|A) = \displaystyle\frac{P(A \cap B)}{P(A)} = \frac{P(B)}{P(A)}\), which is 1 only when P(A) = P(B).
Option (3) \(P(B|A) = 0\) — Incorrect. \(P(B|A) = \displaystyle\frac{P(B)}{P(A)}\), which is not 0 when P(B) > 0.
Option (4) \(P(A|B) = 0\) — Incorrect. \(P(A|B) = \displaystyle\frac{P(A \cap B)}{P(B)} = \frac{P(B)}{P(B)} = 1\), not 0.
NCERT (2026-27 edition):
Probability
