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Question 16

Mathematics Probability Hard

Let U and V be two events of a sample space S and P(A) denote the probability of an event A. Which of the following statements is true?

(A) If P(U) = P(V), then U = V.

(B) If P(U) = 1, then U<sup>c</sup> = S.

(C) If U ∩ V = ϕ, then U and V are independent.

(D) If U and V are independent, then so are U<sup>c</sup> and V<sup>c</sup>.

✓ Correct Answer: Option D Verified Solution

📘 Step-by-Step Solution & Detailed Explanation:

(A) False: equal probabilities don't mean equal events.
(B) False: if P(U) = 1, then \(P(U^c) = 0\). That does not make \(U^c = S\).
(C) False: disjoint events with non-zero probabilities are dependent, since \(P(U \cap V) = 0 \ne P(U)P(V)\).
(D) True: if U and V are independent, then \(P(U^c \cap V^c) = 1 - P(U) - P(V) + P(U)P(V) = (1 - P(U))(1 - P(V))\).

Answer: (D)

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