Question

Let Ω = {1, 2, 3, … } be the sample space of a random experiment and suppose that all subsets of Ω are events. Further, let P be a probability function such that P({i}) > 0 for all i ∈ Ω. Then which of the following statements is/are true ?

• A. For every ∈ > 0, there exists an event A such that 0 < P(A) < ∈
• B. There exists a sequence of disjoint events {Ak}k≥1 with P(Ak) ≥ 10-6 for all k ≥ 1
• C. There exists j ∈ Ω such that P({j}) ≥ P({i}) for all i ∈ Ω
• D. Let {Ak}k≥1 be a sequence of events such that $∑^∞_{k=1} 𝑃(𝐴𝑘) < ∞$. Then for each i ∈ Ω there exists N ≥ 1 (which may depend on 𝑖) such that i ∉ $U^{\infin}_{k=N}A_k$

Answer 1

Answer: (A), (C), (D)

For every ∈ > 0, there exists an event A such that 0 < P(A) < ∈

Answer 2

The correct option are
(A) : For every ∈ > 0, there exists an event A such that 0 < P(A) < ∈,
(C) : There exists j ∈ Ω such that P({j}) ≥ P({i}) for all i ∈ Ω, and
(D) : Let {Ak}k≥1 be a sequence of events such that $∑^∞_{k=1} 𝑃(𝐴𝑘) < ∞$.

Then for each i ∈ Ω there exists N ≥ 1 (which may depend on 𝑖) such that i ∉ $U^{\infin}_{k=N}A_k$