Probability Calculation Selecting Students Not All Boys In A Parade

Jul 16, 2025 by Admin 68 views

Probability of Selecting Students for a Parade Representation

Introduction

In this article, we will delve into the fascinating realm of probability by exploring a problem involving the selection of students for a parade representation. Specifically, we will examine the scenario where two students are randomly chosen from a group comprising eight boys and twelve girls. Our primary objective is to determine the probability that the selected students are not both boys. This problem offers an excellent opportunity to apply fundamental concepts of probability and combinatorics, which are crucial tools for analyzing situations involving chance and randomness. Let's embark on this journey of probability exploration and unravel the solution step by step.

Understanding the Problem

Before diving into the calculations, let's first gain a comprehensive understanding of the problem at hand. We have a total of 20 students, consisting of 8 boys and 12 girls. From this pool of students, we need to select two individuals randomly to represent the school in a parade. The question we aim to answer is: what is the probability that the two students chosen are not both boys? To tackle this problem effectively, we need to consider the different possible outcomes and the specific scenarios that satisfy the condition of not having both students as boys.

Defining Key Terms

To ensure clarity and precision in our analysis, let's define some key terms that will be used throughout this article:

Probability: A measure of the likelihood of an event occurring, expressed as a number between 0 and 1. A probability of 0 indicates that the event is impossible, while a probability of 1 indicates that the event is certain.
Random Selection: The process of choosing individuals from a group in such a way that each individual has an equal chance of being selected.
Combinations: A selection of items from a set where the order of selection does not matter. The number of combinations of n items taken r at a time is denoted as C(n, r) or ⁿCᵣ, and it is calculated as n! / (r!(n-r)!), where "!" represents the factorial function.

Identifying Favorable Outcomes

Now, let's identify the outcomes that satisfy the condition of not having both students as boys. This means we need to consider the following scenarios:

Both students are girls.
One student is a boy, and the other is a girl.

By considering these scenarios, we can ensure that we capture all the possible outcomes where the selected students are not both boys. To calculate the probability of these scenarios, we will employ the principles of combinations and probability.

Calculating the Total Number of Possible Outcomes

To determine the probability of selecting students who are not both boys, we first need to calculate the total number of possible outcomes when choosing two students from the group of 20. This can be calculated using combinations, as the order in which the students are chosen does not matter. The total number of ways to choose 2 students from 20 is given by C(20, 2), which can be calculated as follows:

C(20, 2) = 20! / (2!(20-2)!) = 20! / (2!18!) = (20 * 19) / (2 * 1) = 190

Therefore, there are 190 possible ways to choose two students from the group of 20.

Determining the Number of Unfavorable Outcomes

Next, we need to determine the number of outcomes that are not favorable, meaning the scenarios where both students chosen are boys. This will help us later in calculating the probability of the favorable outcomes. To calculate the number of ways to choose two boys from the group of 8, we again use combinations:

C(8, 2) = 8! / (2!(8-2)!) = 8! / (2!6!) = (8 * 7) / (2 * 1) = 28

So, there are 28 ways to choose two boys from the group of 8.

Calculating the Number of Favorable Outcomes

Now that we know the total number of possible outcomes and the number of unfavorable outcomes (both boys), we can calculate the number of favorable outcomes (not both boys). This can be done by subtracting the number of unfavorable outcomes from the total number of outcomes:

Number of favorable outcomes = Total number of outcomes - Number of unfavorable outcomes Number of favorable outcomes = 190 - 28 = 162

Therefore, there are 162 ways to choose two students such that they are not both boys.

Calculating the Probability

Finally, we can calculate the probability of selecting two students who are not both boys by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability (not both boys) = Number of favorable outcomes / Total number of outcomes Probability (not both boys) = 162 / 190 = 81/95

Thus, the probability that the two students chosen are not both boys is 81/95, which is approximately 0.8526 or 85.26%.

Alternative Approach: Calculating Probabilities Directly

Another way to approach this problem is to calculate the probabilities of the favorable scenarios directly and then add them up. Recall that the favorable scenarios are:

Both students are girls.
One student is a boy, and the other is a girl.

Let's calculate the probabilities for each scenario:

Scenario 1: Both Students are Girls

The number of ways to choose two girls from 12 is C(12, 2) = 12! / (2!10!) = (12 * 11) / (2 * 1) = 66.

The probability of choosing two girls is then the number of ways to choose two girls divided by the total number of ways to choose two students:

Probability (both girls) = 66 / 190

Scenario 2: One Boy and One Girl

The number of ways to choose one boy from 8 is C(8, 1) = 8, and the number of ways to choose one girl from 12 is C(12, 1) = 12. To get one boy and one girl, we multiply these possibilities: 8 * 12 = 96.

The probability of choosing one boy and one girl is the number of ways to choose one boy and one girl divided by the total number of ways to choose two students:

Probability (one boy, one girl) = 96 / 190

Combining the Probabilities

To find the probability that the students chosen are not both boys, we add the probabilities of the two favorable scenarios:

Probability (not both boys) = Probability (both girls) + Probability (one boy, one girl) Probability (not both boys) = (66 / 190) + (96 / 190) = 162 / 190 = 81/95

This approach confirms our earlier result, demonstrating the consistency of the probability calculations.

Conclusion

In conclusion, we have successfully determined the probability that two students chosen at random from a group of eight boys and twelve girls are not both boys. Through the application of combinatorial principles and probability calculations, we found that the probability is 81/95, or approximately 85.26%. This problem highlights the importance of understanding probability concepts and their practical applications in real-world scenarios. By breaking down the problem into smaller steps and carefully considering the different possible outcomes, we were able to arrive at the solution in a clear and logical manner. The use of combinations allowed us to efficiently count the number of ways to select students, while the principles of probability helped us to quantify the likelihood of the desired outcome.

Keywords Explanation

Here are the explanations and fixes for the keywords:

Original Question: If two students are chosen at random from a group of eight boys and twelve girls to represent the school in a parade, what is the probability that the students chosen are not both boys?
Fixed Question: This question is already clear, but it can be rephrased slightly for better understanding: "Two students are selected randomly from a group of 8 boys and 12 girls for a parade representation. What is the probability that the selected students are not both boys?"

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Probability Calculation: Selecting Students Not All Boys in a Parade