Sample Space Coverage: A Probability Question

Hey guys! Today, let's dive into a fun probability problem. Imagine you've got a set of numbers, and you're picking one at random. The question we're tackling is all about understanding what it means for an event to cover the entire sample space. Specifically, we want to figure out which event, by definition, covers the entire sample space of the set {2, 3, 4, ..., 10}. This means we need to identify an event that must occur, no matter which number we pick from that set. Understanding sample spaces and events is fundamental to grasping probability, so let's break it down step by step. We'll explore the different options and see which one guarantees that whatever number we pick from our set will always fit the description of the event. This kind of problem really highlights the importance of carefully defining events and understanding the possible outcomes. Let's get started and unravel this probability puzzle together!

Understanding the Problem

Before we jump into the answer choices, let's make sure we're all on the same page about what the problem is asking. We have a set of numbers: {2, 3, 4, 5, 6, 7, 8, 9, 10}. This is our sample space, which means it's the collection of all possible outcomes of our little experiment (picking a number). An event is just a subset of this sample space, a specific thing that could happen. The question asks us which event completely covers the sample space. In other words, which event must happen, no matter which number we pick. Think of it like this: if we pick any number from 2 to 10, which of the following statements always has to be true? That's what we're trying to find! The concept of a sample space is really the foundation upon which all probability calculations are built. We need to know all the possible outcomes before we can start figuring out the chances of specific events happening. Events, similarly, help us to categorize and analyze the outcomes within the sample space. So, with a firm grasp of these basic ideas, we are now well-equipped to evaluate the answer choices and pinpoint the event that encompasses our entire sample space.

Evaluating the Options

Okay, let's look at some potential answers (which you didn't give me, but let's pretend we have some!). We'll analyze each one to see if it truly covers the entire sample space.

Option A: The number is a positive number not divisible by 5.

Let's think about this. Our set is {2, 3, 4, 5, 6, 7, 8, 9, 10}. If the event is "The number is a positive number not divisible by 5", does this cover everything? No! The numbers 5 and 10 are in our set, and they are divisible by 5. So, if we picked 5 or 10, this event wouldn't happen. Thus, this option doesn't cover the entire sample space. We need an event that always occurs, no matter which number we select. The presence of 5 and 10 immediately disqualifies this option. Understanding divisibility is key here. Remember, a number is divisible by 5 if it can be divided by 5 with no remainder. Since 5 divided by 5 is 1 (no remainder) and 10 divided by 5 is 2 (no remainder), they clearly violate the condition of not being divisible by 5. This careful examination of each number within the set is crucial to determining whether an event truly covers the entire sample space. We can't just make a generalization; we have to confirm that the event holds true for every single element in the set.

Option B: The number is a positive number greater than 2.

Now, what about "The number is a positive number greater than 2"? Looking at our set {2, 3, 4, 5, 6, 7, 8, 9, 10}, we see a problem right away. The number 2 is in the set, but it's not greater than 2. So, if we picked 2, this event wouldn't happen. Therefore, this option also doesn't cover the entire sample space. The definition of 'greater than' is important. A number is greater than 2 if it's strictly larger than 2. Since 2 is equal to itself, it doesn't meet this criterion. This is a good reminder to pay close attention to the wording of the event and ensure that it accurately describes the condition we're testing. The phrase 'greater than' has a very specific mathematical meaning, and we need to apply it rigorously when evaluating whether an event covers the sample space. In probability problems, precision in both the definition of the event and its application to the sample space is absolutely essential for arriving at the correct answer.

Option C: The number is...

Let's consider a hypothetical Option C: "The number is an integer between 1 and 11 inclusive." Inclusive means it includes the endpoints (1 and 11). So, is every number in our set {2, 3, 4, 5, 6, 7, 8, 9, 10} an integer between 1 and 11 inclusive? Yes! 2 is between 1 and 11, 3 is, 4 is, and so on, all the way to 10. So, this event does cover the entire sample space! That's what we're looking for. This option works because it sets a broad boundary that encompasses all the elements in our set. The term 'inclusive' is crucial here because it ensures that the endpoints of the range (1 and 11) are included in the set of valid numbers. If the option had said 'between 1 and 11 exclusive' (meaning not including 1 and 11), then it wouldn't have covered the entire sample space because it would have excluded the number 10. Understanding the nuances of mathematical language is vital when working with sample spaces and events. Small changes in wording can drastically alter the meaning and impact whether an event fully covers the possible outcomes.

The Key Takeaway

The most important thing to remember is that for an event to cover the entire sample space, it must be true for every single possible outcome in the set. There can't be any exceptions. If you find even one number in the set that doesn't fit the event's description, then the event doesn't cover the entire sample space. When faced with these kinds of problems, take each option and carefully check it against every element in your sample space. This methodical approach will help you avoid common mistakes and confidently identify the event that encompasses all possible outcomes. Sample space coverage is a fundamental concept in probability, and mastering it will give you a solid foundation for understanding more advanced topics. By carefully considering all possible outcomes and scrutinizing the definitions of events, you can confidently navigate the world of probability and solve even the most challenging problems.

In Summary

So, to recap, we looked at a problem where we needed to find an event that covers the entire sample space of the set {2, 3, 4, ..., 10}. We learned that an event covers the entire sample space if it always happens, no matter which number we pick from the set. We analyzed some example options and saw how important it is to carefully check each one against every number in the set. If even one number doesn't fit the description of the event, then it doesn't cover the entire sample space. Remember to pay close attention to the wording of the events and understand the mathematical definitions of terms like "greater than" and "inclusive." By practicing these kinds of problems, you'll build a solid understanding of sample spaces and events, which are essential for success in probability. Keep practicing, and you'll become a probability pro in no time! You've got this! Now go out there and conquer those probability problems, guys!