Determining Sample Space For Combinations Carla's Sneaker Selection Problem

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When dealing with probability and combinatorics, understanding the concept of a sample space is crucial. The sample space, often denoted as S, represents the set of all possible outcomes of a particular event or experiment. In this article, we will explore how to determine the sample space in a specific scenario: Carla choosing two pairs of sneakers out of three. We will break down the problem, discuss the fundamental principles involved, and illustrate how to correctly identify the sample space.

Defining the Sample Space

In combinatorics, the sample space is the bedrock upon which all probability calculations are built. It is a comprehensive list of every possible outcome that could occur. Without a clear understanding of the sample space, it becomes impossible to accurately assess probabilities or make predictions. For simple scenarios, listing the sample space might seem straightforward, but as the complexity of the event increases, a systematic approach becomes essential. Consider a simple example, like flipping a coin. The sample space is {Heads, Tails}, representing the two possible outcomes. When we move to more complex scenarios, such as drawing cards from a deck or, as in our case, selecting items from a set, the process of defining the sample space requires careful consideration.

In our specific problem, Carla has three pairs of sneakers, labeled A, B, and C, and she needs to choose two pairs to take to a track meet. The question asks us to identify the sample space S for this event. This means we need to list all possible combinations of two pairs of sneakers that Carla can select. It’s not just about listing the individual pairs but about listing the different ways she can combine them. This is a classic combinatorics problem where order does not matter. The act of selecting pair A and pair B is the same as selecting pair B and pair A for the purposes of this problem. Therefore, we are looking for combinations, not permutations. This is a key distinction in combinatorics that significantly impacts how we approach the problem.

Identifying Possible Combinations

To accurately determine the sample space for Carla's sneaker selection, we need to systematically identify all possible combinations of two pairs she can choose from her three pairs (A, B, and C). A systematic approach ensures that we don't miss any possibilities and avoid duplication. Let's break down the possible combinations:

  1. Pair A and Pair B: Carla can choose to take pairs A and B. This is one possible outcome.
  2. Pair A and Pair C: Carla can choose to take pairs A and C. This is another distinct outcome.
  3. Pair B and Pair C: Carla can choose to take pairs B and C. This is the final possible combination.

It's crucial to recognize that the order in which Carla chooses the pairs doesn't matter. Choosing A and then B is the same as choosing B and then A in this scenario. We are interested in the final selection of two pairs, not the sequence in which they were chosen. This is why we are dealing with combinations rather than permutations. If the order mattered (for example, if the first pair chosen was for a specific event and the second for another), we would have a different problem to solve, one involving permutations. Now that we have identified all the unique pairs, we can represent the sample space concisely.

The sample space, S, is therefore the set of these combinations: {AB, AC, BC}. Each element in this set represents a unique outcome of Carla's sneaker selection process. This sample space forms the basis for calculating probabilities related to this event. For example, if we wanted to know the probability of Carla choosing pair A, we would look at how many elements in the sample space include pair A and divide that by the total number of elements in the sample space.

Common Mistakes to Avoid

When determining sample spaces, particularly in combinatorics problems, there are several common mistakes that students often make. Recognizing and avoiding these pitfalls is essential for accurately solving problems. One frequent mistake is including the same combination in different orders. As we discussed earlier, choosing A and then B is the same outcome as choosing B and then A when we are only concerned with the final selection. Listing both AB and BA would be incorrect because it duplicates the same outcome. Another common error is missing possible combinations. This often happens when students try to list combinations haphazardly rather than using a systematic approach. By methodically pairing each item with every other item, we can ensure that no combination is overlooked.

Another mistake involves confusing combinations and permutations. Combinations are selections where order doesn't matter, while permutations are arrangements where order does matter. The formulas and methods for calculating these are different, so it’s crucial to identify whether a problem involves combinations or permutations before attempting to solve it. In Carla's case, the order in which she chooses the sneakers is irrelevant; only the final pair of sneakers she selects matters. Therefore, this is a combination problem. Finally, it’s important to clearly define the event for which you are constructing the sample space. A poorly defined event can lead to an incorrect sample space. In our case, the event is Carla choosing two pairs of sneakers out of three. The sample space must reflect all possible outcomes of this specific event.

The Correct Sample Space

Based on our systematic analysis, the correct sample space, S, for the event of Carla choosing two pairs of sneakers from three pairs (A, B, and C) is:

S = {AB, AC, BC}

This set includes all possible combinations of two pairs of sneakers that Carla can choose. Each element in the set represents a distinct outcome, and there are no duplicate outcomes. The set accurately reflects the event described in the problem statement. It’s worth noting that none of the originally proposed options—S = ABC} or S = {ABC, Discussion category mathematics—correctly represent the sample space. The first option only lists the three individual pairs, not the combinations of two pairs. The second option is nonsensical as it includes extraneous information (“Discussion category : mathematics”) that has nothing to do with the outcomes of the event. Understanding why these options are incorrect is as important as knowing the correct answer.

Application of Combinatorics Principles

The problem of determining the sample space for Carla's sneaker selection is a practical application of combinatorics principles. Combinatorics is a branch of mathematics concerned with counting, arrangement, and combination of objects. It provides the tools and techniques necessary to solve problems involving discrete structures and finite sets. Understanding these principles allows us to tackle a wide range of problems, from simple selection tasks to complex probability calculations.

The number of ways to choose k items from a set of n items without regard to order is given by the combination formula:

C(n, k) = n! / (k!(n-k)!)

where "!" denotes the factorial function (e.g., 5! = 5 × 4 × 3 × 2 × 1). In Carla's case, we want to choose 2 pairs of sneakers from a set of 3 pairs. So, n = 3 and k = 2. Applying the formula:

C(3, 2) = 3! / (2!(3-2)!) = 3! / (2!1!) = (3 × 2 × 1) / (2 × 1 × 1) = 3

This confirms that there are 3 possible combinations, which aligns with our manually derived sample space {AB, AC, BC}. The combination formula provides a quick and efficient way to calculate the number of combinations without having to list them all out, especially when dealing with larger sets.

Combinatorics principles are not only useful in mathematics but also have applications in various fields, including computer science, statistics, physics, and even economics. For instance, in computer science, combinatorics is used in algorithm design and analysis. In statistics, it is essential for probability calculations and experimental design. The ability to understand and apply these principles is a valuable skill in many areas of study and work.

Conclusion

In summary, accurately determining the sample space is a fundamental step in solving combinatorics and probability problems. By systematically identifying all possible outcomes and avoiding common mistakes, we can construct the sample space correctly. In the case of Carla choosing two pairs of sneakers from three, the sample space is S = {AB, AC, BC}. This set represents all possible combinations of two pairs that Carla can select. Understanding the underlying principles of combinatorics, such as combinations versus permutations, is crucial for solving these types of problems. Furthermore, the ability to apply the combination formula provides a powerful tool for calculating the number of combinations in more complex scenarios. This article has provided a detailed explanation of how to determine the sample space in this specific scenario, highlighting the importance of a systematic approach and the avoidance of common errors. By mastering these concepts, students and practitioners alike can confidently tackle a wide range of combinatorics and probability problems.