Which Event, By Definition, Covers The Entire Sample Space Of This Experiment?

In the realm of probability, grasping the concept of sample space is fundamental. The sample space encompasses all possible outcomes of a random experiment. Let's delve into a problem where we need to identify the event that covers the entire sample space.

Problem Statement: Selecting a Number at Random

Consider the scenario where a number is selected at random from the set ${{2, 3, 4, \ldots, 10}}$. Our task is to determine which event, by definition, covers the entire sample space of this experiment. The options presented are:

A. The number is a positive number greater than 2. B. The number is a...

To solve this problem effectively, we need to meticulously analyze each option and assess whether it encompasses all possible outcomes within the given sample space. The sample space in this case is the set of numbers from 2 to 10, inclusive. Therefore, any event that includes all these numbers will be the one that covers the entire sample space. Let's break down the options to identify the correct one.

Analyzing Option A: The Number is a Positive Number Greater Than 2

This option states that the selected number is a positive number greater than 2. While all the numbers in our sample space ${{2, 3, 4, \ldots, 10}}$ are indeed greater than 2, this statement doesn't quite capture the entire sample space. The reason is that the sample space explicitly starts from 2. While the numbers 3, 4, 5, 6, 7, 8, 9, and 10 fit this description, the number 2 itself is excluded by the phrase "greater than 2." Therefore, option A does not cover the entire sample space because it omits the number 2, which is a valid outcome in our experiment. To fully cover the sample space, the event must include all possible outcomes, and in this case, 2 is a crucial element.

Furthermore, it's essential to understand the nuances of mathematical language. The phrase "greater than 2" strictly means numbers that are larger than 2, excluding 2 itself. If the phrase were "greater than or equal to 2," then 2 would be included. This subtle distinction is vital in probability and set theory, where precision is paramount. The inclusion or exclusion of a single element can significantly alter the outcome and the interpretation of an event. Thus, when evaluating events against a sample space, it's crucial to pay attention to the exact wording and its mathematical implications.

In summary, while option A encompasses a significant portion of the sample space, its exclusion of the number 2 means it falls short of covering the entire sample space. This highlights the importance of a comprehensive understanding of the sample space and the precise definition of events when solving probability problems. The event must unequivocally include every possible outcome to be considered as covering the entire sample space.

[To be continued with the analysis of other options and the final answer]

Continuing the Analysis: Identifying the Correct Event

To correctly identify the event that covers the entire sample space, we need to consider what constitutes a complete and exhaustive description of all possible outcomes. As previously established, the sample space is the set ${{2, 3, 4, \ldots, 10}}$. This means any event that includes every number from 2 to 10 will cover the sample space. We've already examined option A and found it lacking because it excludes the number 2. Now, let's consider other potential options that might fully encompass the sample space.

Hypothetical Option B: The Number is an Integer Between 2 and 10, Inclusive

Let's introduce a hypothetical option B: "The number is an integer between 2 and 10, inclusive." This option is carefully worded to include both 2 and 10, as well as all the integers in between. The term "inclusive" is crucial here, as it explicitly states that the endpoints (2 and 10) are part of the set. This option accurately describes the sample space because it covers every possible outcome of the experiment. When a number is selected at random from the set ${{2, 3, 4, \ldots, 10}}$, the result will invariably be an integer between 2 and 10, inclusive. Therefore, this event covers the entire sample space by definition.

To further illustrate why this option is correct, consider each number in the sample space individually. The number 2 is an integer between 2 and 10, inclusive. The number 3 is also an integer between 2 and 10, inclusive. This pattern continues for all numbers up to 10. Thus, every element in the sample space is accounted for by this event. The use of "inclusive" is the key factor here, distinguishing this option from option A, which excluded 2.

The Importance of "Inclusive" in Mathematical Definitions

The word "inclusive" is a cornerstone of mathematical precision. It ensures that the boundaries of a range or set are included in the consideration. Without "inclusive," the meaning can shift, leading to misinterpretations and incorrect conclusions. In the context of sample spaces and events, this precision is essential. An event must unequivocally include all possible outcomes to cover the entire sample space. The subtle difference between "greater than 2" and "greater than or equal to 2" (or "between 2 and 10" and "between 2 and 10, inclusive") highlights this importance.

In summary, a well-defined event that covers the entire sample space must leave no room for ambiguity. It should explicitly include all possible outcomes, ensuring that no element of the sample space is omitted. The hypothetical option B, "The number is an integer between 2 and 10, inclusive," serves as an example of such an event, perfectly encapsulating the entire sample space of the experiment.

Determining the Correct Answer and Final Thoughts

In the context of the original problem, we were presented with a scenario where a number is selected at random from the set ${{2, 3, 4, \ldots, 10}}$. We analyzed option A, "The number is a positive number greater than 2," and found it to be insufficient because it excludes the number 2. The hypothetical option B, "The number is an integer between 2 and 10, inclusive," was identified as a potential correct answer because it covers the entire sample space.

To arrive at the definitive answer, we must consider all provided options and evaluate each against the definition of the sample space. A correct event, by definition, includes every possible outcome of the experiment. This means that the event must encompass all numbers from 2 to 10, without omission.

Upon careful consideration, the event that accurately covers the sample space is one that explicitly states the inclusion of all numbers within the range. For instance, an event phrased as "The number is an integer from 2 to 10, inclusive" would be correct. This phrasing leaves no ambiguity and ensures that every element of the sample space is accounted for.

Key Takeaways

1. Sample Space Definition: The sample space is the set of all possible outcomes of an experiment. Understanding the sample space is crucial for solving probability problems.
2. Event Coverage: An event covers the entire sample space if it includes all possible outcomes. No outcome should be omitted.
3. Mathematical Precision: The language used to define events must be precise. Words like "inclusive" have significant mathematical implications.
4. Comprehensive Analysis: Each option must be thoroughly analyzed to determine if it fully covers the sample space.

In conclusion, identifying the event that covers the entire sample space requires a clear understanding of what constitutes a complete description of all possible outcomes. Precision in language and a comprehensive analysis of options are essential tools in solving such problems.

#seo title: Probability Sample Space Identifying the Covering Event