Probability Of Selecting Two Sophomores For Debate Team Alternates

JU07/23/2025 08, 2025 by THE IDEN 67 views

Introduction

In this article, we will delve into the world of probability, focusing on a specific scenario involving the selection of debate team alternates. We are presented with a situation where six sophomores and fourteen freshmen are vying for two alternate positions. Our primary goal is to determine the probability that both students chosen for these coveted positions are sophomores. This problem requires us to understand and apply the concepts of combinations, which are a fundamental aspect of probability theory. We will explore the underlying principles of combinations, learn how to calculate them, and then apply this knowledge to solve the given problem. Understanding the probability of such events is crucial in various real-world scenarios, from selecting a team to conducting scientific experiments. So, let's embark on this journey to unravel the mysteries of probability and find the solution to our intriguing problem.

Understanding Combinations

At the heart of our probability problem lies the concept of combinations. In mathematics, a combination is a selection of items from a set where the order of selection does not matter. This is in contrast to permutations, where the order is crucial. For example, if we are choosing two students from a group of four, the combination of students A and B is the same as the combination of students B and A. The formula for calculating combinations is given by:

$\qquad C(n, k) = \frac{n!}{k!(n-k)!}$

Where:

$n$ is the total number of items in the set
$k$ is the number of items to be chosen
$n!$ (n factorial) is the product of all positive integers up to n (e.g., 5! = 5 × 4 × 3 × 2 × 1)

The combination formula helps us determine the number of ways to choose a subset of items from a larger set without regard to order. This is particularly useful in probability problems where we want to find the likelihood of a specific subset being selected. In our debate team problem, we need to calculate the number of ways to choose two sophomores from the group of six, as well as the total number of ways to choose two students from the combined group of sophomores and freshmen. By understanding and applying the combination formula, we can accurately determine the probabilities involved in this scenario.

Applying Combinations to the Debate Team Problem

Now, let's apply our understanding of combinations to the debate team problem. We have six sophomores and fourteen freshmen competing for two alternate positions. We want to find the probability that both students chosen are sophomores. To do this, we need to calculate two things:

The number of ways to choose two sophomores from the six available.
The total number of ways to choose two students from the entire group of twenty (six sophomores plus fourteen freshmen).

Calculating the Number of Ways to Choose Two Sophomores

We can use the combination formula to find the number of ways to choose two sophomores from six. Here, $n = 6$ (the total number of sophomores) and $k = 2$ (the number of sophomores we want to choose). Plugging these values into the formula, we get:

$\qquad C(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6!}{2!4!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(4 \times 3 \times 2 \times 1)} = \frac{6 \times 5}{2 \times 1} = 15$

So, there are 15 different ways to choose two sophomores from the six available.

Calculating the Total Number of Ways to Choose Two Students

Next, we need to find the total number of ways to choose two students from the entire group of twenty. In this case, $n = 20$ (the total number of students) and $k = 2$ (the number of students we want to choose). Using the combination formula again:

$\qquad C(20, 2) = \frac{20!}{2!(20-2)!} = \frac{20!}{2!18!} = \frac{20 \times 19 \times 18!}{(2 \times 1)(18!)} = \frac{20 \times 19}{2 \times 1} = 190$

Therefore, there are 190 different ways to choose two students from the entire group of twenty.

Calculating the Probability

Now that we have calculated the number of ways to choose two sophomores and the total number of ways to choose two students, we can determine the probability of selecting two sophomores. The probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes. In this case:

Favorable outcomes: Choosing two sophomores (15 ways)
Total possible outcomes: Choosing any two students (190 ways)

So, the probability of choosing two sophomores is:

$\qquad P(\text{two sophomores}) = \frac{\text{Number of ways to choose two sophomores}}{\text{Total number of ways to choose two students}} = \frac{15}{190}$

We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

$\qquad P(\text{two sophomores}) = \frac{15 \div 5}{190 \div 5} = \frac{3}{38}$

Therefore, the probability that both students chosen are sophomores is $\frac{3}{38}$ .

The Correct Expression

Now, let's consider the expressions provided in the original problem. We are looking for an expression that represents the probability we just calculated, which is $\frac{15}{190}$ or $\frac{3}{38}$ . The expression that correctly represents this probability is:

$\qquad \frac{{ }_6 C_2}{{ }_{20} C_2}$

This expression directly corresponds to our calculations:

${ }_6 C_2$ represents the number of ways to choose 2 sophomores from 6, which we calculated as 15.
${ }_{20} C_2$ represents the total number of ways to choose 2 students from 20, which we calculated as 190.

Therefore, the expression $\frac{{ }_6 C_2}{{ }_{20} C_2}$ accurately represents the probability that both students chosen are sophomores.

Distinguishing Combinations from Permutations

It's crucial to understand the difference between combinations and permutations, as using the wrong one can lead to incorrect probability calculations. As we discussed earlier, combinations are used when the order of selection does not matter, while permutations are used when the order is important. To further illustrate this distinction, let's consider our debate team problem.

If we were selecting two students for specific roles, such as a lead speaker and a second speaker, the order would matter. Choosing sophomore A as the lead speaker and sophomore B as the second speaker is different from choosing sophomore B as the lead speaker and sophomore A as the second speaker. In this case, we would use permutations.

The formula for permutations is:

$\qquad P(n, k) = \frac{n!}{(n-k)!}$

Where:

$n$ is the total number of items in the set
$k$ is the number of items to be chosen

However, in our problem, the two alternate positions are not distinct, meaning that the order in which the students are chosen does not matter. Therefore, combinations are the appropriate tool for calculating the probabilities involved. Understanding this distinction is essential for accurately solving probability problems.

Real-World Applications of Combinations and Probability

The concepts of combinations and probability are not just theoretical exercises; they have numerous real-world applications across various fields. Let's explore some examples:

Lotteries: Calculating the probability of winning a lottery involves understanding combinations. The lottery selects a certain number of winning numbers from a larger set, and the order in which the numbers are drawn does not matter. Therefore, combinations are used to determine the odds of winning.
Card Games: Many card games, such as poker and bridge, involve calculating the probability of drawing specific hands. These calculations rely heavily on combinations, as the order in which the cards are dealt is irrelevant.
Quality Control: In manufacturing, combinations are used to select samples for quality control testing. A certain number of items are randomly chosen from a batch, and the probability of selecting defective items can be calculated using combinations.
Genetics: Combinations are used in genetics to determine the probability of inheriting specific traits. Genes are inherited in pairs, and the possible combinations of genes from each parent determine the traits of the offspring.
Sampling and Surveys: When conducting surveys or research studies, combinations are used to select a representative sample from a larger population. This ensures that the sample is random and unbiased.

These are just a few examples of how combinations and probability are used in real-world scenarios. The ability to understand and apply these concepts is valuable in many different fields.

Conclusion

In this article, we have explored the concept of combinations and its application to a probability problem involving the selection of debate team alternates. We learned how to calculate combinations using the formula $C(n, k) = \frac{n!}{k!(n-k)!}$ and applied this knowledge to determine the probability that both students chosen for the alternate positions are sophomores. We also highlighted the importance of distinguishing between combinations and permutations, as well as the real-world applications of these concepts.

The problem we solved demonstrates the power of probability theory in analyzing and understanding random events. By breaking down the problem into smaller steps and applying the appropriate formulas, we were able to arrive at the correct solution. Understanding combinations and probability is not only essential for solving mathematical problems but also for making informed decisions in various aspects of life. From assessing risks to making predictions, the principles of probability provide a valuable framework for navigating the uncertainties of the world around us. Therefore, mastering these concepts is a worthwhile endeavor for anyone seeking to enhance their analytical and problem-solving skills.