Question

Let \( \Omega = \{1, 2, 3, \dots \} \) represent the collection of all possible outcomes of a random experiment with probabilities \( P(\{n\}) = a_n \) for \( n \in \Omega \). Then which one of the following statements is NOT true?

• A. \( \lim_{n \to \infty} a_n = 0 \)
• B. \( \sum_{n=1}^{\infty} \sqrt{a_n} \) converges
• C. For any positive integer \( k \), there exist \( k \) disjoint events \( A_1, A_2, \dots, A_k \) such that \( P\left( \bigcup_{i=1}^{k} A_i \right)<0.001 \)
• D. There exists a sequence \( \{A_i\}_{i \geq 1} \) of strictly increasing events such that \( P\left( \bigcup_{i=1}^{\infty} A_i \right)<0.001 \)

Hint:

For infinite series of probabilities, check the behavior of individual terms and ensure that the series converges based on the given conditions.

Answer 1

The Correct Option is B

We are given a random experiment with probabilities \( P(\{n\}) = a_n \) for \( n \in \Omega \). The statements deal with the convergence of sums of probabilities and events. Step 1: Analyze each option.
- Option (A) is correct because for a probability distribution, the individual probabilities \( a_n \) must tend to zero as \( n \to \infty \).
- Option (B) is incorrect because the series \( \sum_{n=1}^{\infty} \sqrt{a_n} \) cannot always be assumed to converge for all probability distributions.
- Option (C) is correct because for any \( k \), we can always find disjoint events whose union has a probability less than 0.001, given that the probabilities \( a_n \) are small.
- Option (D) is correct because it is possible to construct a sequence of events such that the probability of their union is less than 0.001.
Final Answer: \[ \boxed{\sum_{n=1}^{\infty} \sqrt{a_n} \, \text{does not always converge}}. \]