Probability Of Selecting Students Not Both Boys For A Parade Representation

In this article, we will delve into the fascinating world of probability, specifically addressing a problem involving the selection of students for a school parade. Probability, a cornerstone of mathematics and statistics, quantifies the likelihood of an event occurring. It plays a crucial role in diverse fields, from predicting weather patterns to assessing financial risks. Understanding probability allows us to make informed decisions in the face of uncertainty.

In this particular scenario, we have a group of eight boys and 12 girls, and two students are randomly chosen to represent the school in a parade. Our primary objective is to determine the probability that the selected students are not both boys. This involves calculating the probability of the complementary event – the probability that both students are boys – and subtracting it from 1. By systematically breaking down the problem and applying fundamental probability principles, we will arrive at the solution.

This problem not only tests our understanding of probability but also highlights the importance of careful analysis and logical reasoning. We must consider all possible outcomes and identify the specific events that satisfy the given condition. Furthermore, we will explore the concepts of combinations, which are essential for counting the number of ways to select a group of items from a larger set without regard to order. By the end of this discussion, you will have a clear understanding of how to approach similar probability problems and appreciate the power of mathematical tools in solving real-world scenarios.

Probability forms the bedrock of our exploration, specifically, the probability of selecting students for a parade. Probability is fundamentally the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In simpler terms, it tells us how likely something is to happen. The formula for calculating probability is straightforward:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes)

Understanding this basic formula is crucial for tackling probability problems. The "number of favorable outcomes" refers to the specific outcomes that we are interested in, while the "total number of possible outcomes" represents all the possible scenarios that could occur. For instance, if we flip a fair coin, there are two possible outcomes (heads or tails), and the probability of getting heads is 1/2 because there is one favorable outcome (heads) out of two possible outcomes.

In our student selection scenario, the problem statement lays out the specifics: we have a group of eight boys and 12 girls, totaling 20 students. Two students are chosen at random to represent the school in a parade. The core question we aim to answer is: what is the probability that the students chosen are not both boys? This seemingly simple question requires us to delve into the principles of combinatorics and probability to arrive at the correct answer.

The phrase "not both boys" is key here. It implies that we are interested in all possible combinations of students except the case where both students selected are boys. This sets the stage for us to either directly calculate the probabilities of the favorable outcomes (one boy and one girl, or two girls) or to use the concept of complementary probability, which often simplifies calculations in such scenarios. We'll explore both approaches in the following sections, but first, let's dissect the total possible outcomes.

To determine the total possible outcomes, we delve into the realm of combinatorics, specifically combinations. Combinations are a way of selecting items from a set where the order of selection does not matter. In our case, we are selecting two students from a group of 20, and the order in which they are chosen is irrelevant. This is a classic combination problem.

The formula for calculating combinations is expressed as:

nCr = n! / (r! * (n-r)!)

Where:

• n is the total number of items in the set
• r is the number of items being chosen
• ! denotes the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1)

Applying this formula to our problem, we have n = 20 (total students) and r = 2 (students being chosen). Thus, the total number of ways to choose two students from 20 is:

20C2 = 20! / (2! * 18!)

Let's break this down step by step. 20! means 20 * 19 * 18 * ... * 2 * 1, and 18! means 18 * 17 * ... * 2 * 1. We can simplify the expression by canceling out the common terms in the numerator and denominator:

20C2 = (20 * 19 * 18!) / (2 * 1 * 18!)

20C2 = (20 * 19) / (2 * 1)

20C2 = 380 / 2

20C2 = 190

Therefore, there are 190 different ways to choose two students from the group of 20. This number represents the denominator in our probability calculations – the total number of possible outcomes. Now that we've established this crucial piece of information, we can move on to calculating the number of unfavorable outcomes, which in our case, is the scenario where both selected students are boys.

Now, let's focus on the unfavorable outcomes: the scenario where both selected students are boys. To calculate this, we again use the combination formula, but this time, we are selecting two boys from the group of eight boys. So, n = 8 (total boys) and r = 2 (boys being chosen).

Applying the combination formula:

8C2 = 8! / (2! * 6!)

Expanding the factorials and simplifying:

8C2 = (8 * 7 * 6!) / (2 * 1 * 6!)

8C2 = (8 * 7) / (2 * 1)

8C2 = 56 / 2

8C2 = 28

This tells us that there are 28 ways to choose two boys from the group of eight. These 28 combinations represent the outcomes we want to exclude when calculating the probability that the students chosen are not both boys. They form the basis for calculating the probability of the complementary event – the probability that both students are boys.

Understanding the number of unfavorable outcomes is crucial because it allows us to use the principle of complementary probability. The probability of an event not happening is equal to 1 minus the probability of the event happening. In our case, the event is