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 {A_k}_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

The Correct Option is A, C, D

Let's analyze each of the given statements to determine which ones are true.

1. Statement: For every \(∈ > 0\), there exists an event \(A\) such that \(0 < P(A) < ∈\). 
   Analysis: Since \(Ω = \{1, 2, 3, ...\}\) is an infinite sample space, and each singleton event has positive probability \(P(\{i\})\), we can always find a subset \(A\) with small enough probability that fits any specified upper bound \(∈\). This condition satisfies the requirement of countable additivity and the nature of infinite series. 
   Conclusion: True.
2. Statement: There exists a sequence of disjoint events \(\{A_k\}_{k \geq 1}\) with \(P(A_k) \geq 10^{-6}\) for all \(k \geq 1\). 
   Analysis: If such sequence existed, the sum of their probabilities \(\sum_{k=1}^{\infty} P(A_k)\) would be infinite since each \(P(A_k) \geq 10^{-6}\), contradicting the constraint that the probability over the entirety of the infinite sum should be constrained by the total probability not exceeding 1. 
   Conclusion: False.
3. Statement: There exists \(j ∈ Ω\) such that \(P(\{j\}) \geq P(\{i\})\) for all \(i ∈ Ω\). 
   Analysis: Given an infinite sequence of positive numbers, there exists at least one maximal value (i.e., a number which is not less than any other number in the sequence). Thus, such a \(j\) must exist in the countable sample space where probabilities of singleton sets are positive numbers. 
   Conclusion: True.
4. Statement: Let \(\{A_k\}_{k \geq 1}\) be a sequence of events such that \(\sum_{k=1}^{\infty} P(A_k) < \infty\). Then for each \(i ∈ Ω\), there exists \(N \geq 1\) (which may depend on \(i\)) such that \(i \notin \bigcup_{k=N}^{\infty} A_k\). 
   Analysis: This relates to the Borel-Cantelli Lemma criterion considering \(\sum_{k=1}^{\infty} P(A_k) < \infty\). For each \(i\), if an event occurs only finitely often with positive overall measure less than infinity, eventually \(i\) will not appear in the continued series from some point \(N\) onward. 
   Conclusion: True.

Therefore, the correct answers are:

• For every \(∈ > 0\), there exists an event \(A\) such that \(0 < P(A) < ∈\)
• There exists \(j ∈ Ω\) such that \(P(\{j\}) \geq P(\{i\})\) for all \(i ∈ Ω\)
• Let \(\{A_k\}_{k \geq 1}\) be a sequence of events such that \(\sum_{k=1}^{\infty} P(A_k) < \infty\). Then for each \(i ∈ Ω\), there exists \(N \geq 1\) such that \(i \notin \bigcup_{k=N}^{\infty} A_k\)