For any events A and B of a sample space S, which of the following statements are TRUE?
(A) $P(S|B) = 1$
(B) $P(A ∩ B) = P(A) + P(B) + P(A ∪B)$
(C) $P(\bar A|B) = 1-P(A|B)$
(D) $P(A|B)=\frac{P(A∩B)}{P(B)}, P(B) ≠ 0$
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (C) and (D) only **
Check each statement using basic probability rules.
(A)
$P(S\mid B)=1$
Since $S$ is the entire sample space, conditioning on $B$ does not change it. Always true.
(B)
$P(A\cap B)=P(A)+P(B)+P(A\cup B)$
This is incorrect because the correct identity is:
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$
(C)
$P(\bar A\mid B)=1-P(A\mid B)$
This is true because:
$P(\bar A\mid B)=\frac{P(B)-P(A\cap B)}{P(B)}=1-\frac{P(A\cap B)}{P(B)}$
$=1-P(A\mid B)$
(D)
$P(A\mid B)=\frac{P(A\cap B)}{P(B)},\;P(B)\neq0$
This is the standard definition of conditional probability → True.
Correct statements: (A), (C), (D)