**Which statement(s) is/are true?** *This is a multiple-answers question. Multiple statements can be true.*

The probability of a particular outcome in a randomized experiment can be defined as the proportion of times this outcome occurs in the long run, when this experiment is repeated over and over.

With random phenomena, the proportion of times that something happens is highly random and variable in the short run, but very predictable in the long run.

Even with independent trials, the probability of a particular outcome can fluctuate with every repetition of an experiment.

Let's evaluate each statement in detail:

1. **The probability of a particular outcome in a randomized experiment can be defined as the proportion of times this outcome occurs in the long run, when this experiment is repeated over and over.**
   - This statement is **true**. This is a classic definition of probability known as the *frequentist* interpretation. It suggests that the probability of an event is the long-run relative frequency of occurrence of that event in repeated independent trials.

2. **With random phenomena, the proportion of times that something happens is highly random and variable in the short run, but very predictable in the long run.**
   - This statement is **true**. In the short run, random variation can lead to unpredictability. However, in the long run, the Law of Large Numbers states that the proportion of occurrences of an event will converge to the true probability, making it predictable.

3. **Even with independent trials, the probability of a particular outcome can fluctuate with every repetition of an experiment.**
   - This statement is **false**. The probability of a particular outcome is a fixed value for a given experiment (assuming the conditions of the experiment remain unchanged). While the outcomes themselves fluctuate, the probability remains constant across independent trials.

### Conclusion:
- Statements 1 and 2 are **true**.
- Statement 3 is **false**.

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Do you have any questions or want further details? Here are 5 related questions you can explore:

1. What is the Law of Large Numbers, and how does it relate to probability?
2. Can probability theory be applied to non-random phenomena?
3. How does the frequentist interpretation of probability differ from the Bayesian interpretation?
4. What are the key assumptions behind independent trials in probability experiments?
5. How do short-run fluctuations impact our ability to predict outcomes in probability?

**Tip:** In any probability experiment, larger sample sizes tend to yield results closer to the true probability due to the Law of Large Numbers.

Which statement(s) is/are true? This is a multiple-answers question. Multiple statements can be true.

The probability of a particular outcome in a randomized experiment can be defined as the proportion of times this outcome occurs in the long run, when this experiment is repeated over and over.

With random phenomena, the proportion of times that something happens is highly random and variable in the short run, but very predictable in the long run.

Even with independent trials, the probability of a particular outcome can fluctuate with every repetition of an experiment.

Explore the truth of multiple statements regarding probability theory. This problem involves key concepts such as the frequentist interpretation of probability, the Law of Large Numbers, and short-run variability in random phenomena. Suitable for college-level students.

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