Probability Of Selecting A Scientist, Broadcaster, And Artist

In probability theory, we often encounter scenarios where we need to determine the likelihood of specific events occurring when selecting items from a group. This article delves into a practical problem involving a class of 45 students with varying talents in Mathematics and Science, English Language, and Crafts. Our objective is to calculate the probability of selecting a student who excels in science (Scientist), a student with excellent communication skills (Broadcaster), and a student with artistic talent (Artist) on graduation day. To solve this problem, we will leverage fundamental concepts of probability, specifically the principles of combinations and conditional probability.

To effectively tackle this probability question, let's dissect the given information and outline the steps required to arrive at the solution. First, we recognize that we're dealing with a finite set of students, each possessing different skill sets. The challenge lies in figuring out the likelihood of picking a specific combination of students—one representing science, one representing communication, and one representing artistic ability—from the overall group. This involves not only understanding the individual probabilities of selecting a student from each category but also how these probabilities interact when making multiple selections. We will use a step-by-step approach, breaking down the problem into smaller, more manageable parts to ensure clarity and accuracy in our calculations. The approach will highlight how each selection affects the remaining pool of students and the overall probability.

Before we dive into the calculations, let's clearly define the categories and the number of students who fall into each:

• Scientists: This category includes students who are proficient in Mathematics and Science. According to the problem, there are 10 such students.
• Broadcasters: This category comprises students who excel in the English Language. There are 15 students in this group.
• Artists: This category consists of students who are skilled in Crafts. We have 20 students in this category.

It's important to note that some students may belong to multiple categories, which adds a layer of complexity to our calculations. We will address this by considering the overlaps when we compute the probabilities. This initial categorization is crucial because it sets the stage for calculating the probabilities of selecting a student from each category, which is the foundation of our solution. The precision in this step ensures that subsequent calculations accurately reflect the scenario described in the problem.

Calculating the Probability: A Step-by-Step Approach

To find the probability of selecting a Scientist, a Broadcaster, and an Artist, we will break down the problem into a series of steps, each representing the selection of one type of student. This approach allows us to account for the changing composition of the student pool after each selection, ensuring an accurate final probability.

Step 1: Probability of Selecting a Scientist

The first step is to determine the probability of selecting a Scientist from the class. There are 10 students who excel in Mathematics and Science, and a total of 45 students in the class. The probability of selecting a Scientist on the first pick is therefore:

$$P(Scientist) = \frac{\text{Number of Scientists}}{\text{Total Number of Students}} = \frac{10}{45}$$

This initial calculation sets the stage for the subsequent selections, as it establishes the likelihood of the first event occurring. It's important to note that this probability is based on the initial conditions of the problem, with all 45 students available for selection. The result of this step directly influences the probabilities in the following steps, as the pool of students and the number of students in each category will change.

Step 2: Probability of Selecting a Broadcaster After Selecting a Scientist

After selecting a Scientist, we need to calculate the probability of selecting a Broadcaster. This step introduces the concept of conditional probability, as the probability of selecting a Broadcaster depends on the outcome of the previous selection. Assuming that the student selected in the first step is not also a Broadcaster, we now have 44 students remaining. However, we need to consider the overlap between Scientists and Broadcasters. To simplify, we'll assume the selected Scientist is not also a Broadcaster. There are 15 students who are good in English Language (Broadcasters). Thus, the probability of selecting a Broadcaster after selecting a Scientist is:

$$P(Broadcaster | Scientist) = \frac{\text{Number of Broadcasters}}{\text{Total Remaining Students}} = \frac{15}{44}$$

This step is crucial because it illustrates how the probability changes as selections are made. The conditionality introduced here highlights the interdependence of events in probability calculations. By accounting for the reduced pool of students, we ensure a more accurate reflection of the actual likelihood of selecting a Broadcaster after a Scientist has already been chosen.

Step 3: Probability of Selecting an Artist After Selecting a Scientist and a Broadcaster

Following the selection of a Scientist and a Broadcaster, we now calculate the probability of selecting an Artist. Again, we assume that neither of the previously selected students is an Artist. We have 43 students remaining. There are 20 students skilled in Crafts (Artists). The probability of selecting an Artist after selecting a Scientist and a Broadcaster is:

$$P(Artist | Scientist \cap Broadcaster) = \frac{\text{Number of Artists}}{\text{Total Remaining Students}} = \frac{20}{43}$$

This final conditional probability reflects the culmination of the previous selections and their impact on the remaining student pool. It underscores the importance of considering the sequential nature of the selections and how each choice affects the subsequent probabilities. The accuracy of this step is vital for determining the overall probability of the desired outcome.

Calculating the Overall Probability

To find the overall probability of selecting a Scientist, a Broadcaster, and an Artist in that order, we multiply the probabilities from each step:

$$P(Scientist \cap Broadcaster \cap Artist) = P(Scientist) \times P(Broadcaster | Scientist) \times P(Artist | Scientist \cap Broadcaster)$$

Substituting the values we calculated earlier:

$$P(Scientist \cap Broadcaster \cap Artist) = \frac{10}{45} \times \frac{15}{44} \times \frac{20}{43}$$

Now, let's simplify the expression:

$$P(Scientist \cap Broadcaster \cap Artist) = \frac{10 \times 15 \times 20}{45 \times 44 \times 43} = \frac{3000}{85140}$$

Reducing the fraction to its simplest form:

$$P(Scientist \cap Broadcaster \cap Artist) = \frac{50}{1419}$$

Therefore, the probability of selecting a Scientist, a Broadcaster, and an Artist from the class is approximately 50 out of 1419. This result encapsulates the combined likelihood of all three events occurring in sequence, taking into account the dependencies between them.

It's important to recognize that the order in which we select the students—Scientist, Broadcaster, and Artist—is just one possible sequence. There are other orders in which these selections could occur, and each order would have its own probability. To get the total probability of selecting one student from each category, regardless of the order, we need to consider all possible permutations. This involves understanding how rearranging the order of selections affects the overall likelihood of the desired outcome. The analysis of different selection orders adds a layer of complexity to the problem but provides a more complete understanding of the probabilities involved.

Considering All Permutations

There are 3! (3 factorial) ways to arrange three distinct items, which is 3 x 2 x 1 = 6. These permutations represent all the possible orders in which we could select a Scientist, a Broadcaster, and an Artist. To find the total probability of selecting one student from each category in any order, we need to calculate the probability for each order and sum them up. This approach ensures that we account for every possible sequence of selections, providing a comprehensive view of the overall likelihood.

The Permutations

Let's list the six possible orders:

1. Scientist, Broadcaster, Artist (which we already calculated)
2. Scientist, Artist, Broadcaster
3. Broadcaster, Scientist, Artist
4. Broadcaster, Artist, Scientist
5. Artist, Scientist, Broadcaster
6. Artist, Broadcaster, Scientist

For each of these orders, we would calculate the probability in a similar way to what we did earlier, considering the changing pool of students after each selection. The probabilities for each order may differ slightly due to the conditional nature of the selections. Calculating each permutation allows for a precise determination of the overall probability, as it accounts for the nuanced changes in likelihood as the selection order varies.

Calculating Probabilities for Each Permutation

To illustrate, let's calculate the probability for the order Scientist, Artist, Broadcaster:

$$P(Scientist) = \frac{10}{45}$$

$$P(Artist | Scientist) = \frac{20}{44}$$

$$P(Broadcaster | Scientist \cap Artist) = \frac{15}{43}$$

$$P(Scientist \cap Artist \cap Broadcaster) = \frac{10}{45} \times \frac{20}{44} \times \frac{15}{43} = \frac{3000}{85140} = \frac{50}{1419}$$

Interestingly, this probability is the same as the first order we calculated. This is because the numerators and denominators are simply rearranged, but the product remains the same. This consistency simplifies the overall calculation, as we can leverage this pattern across all permutations. The recognition of this pattern is a crucial insight that streamlines the process of calculating the total probability.

Summing the Probabilities

Since the probability for each permutation is the same (rac{50}{1419}), we can calculate the total probability by multiplying this probability by the number of permutations (6):

$$P(Total) = 6 \times \frac{50}{1419} = \frac{300}{1419}$$

Reducing the fraction:

$$P(Total) = \frac{100}{473}$$

Therefore, the total probability of selecting a Scientist, a Broadcaster, and an Artist in any order is approximately 100 out of 473. This final result provides a comprehensive answer to the problem, accounting for all possible selection sequences and their respective probabilities.

In this article, we explored a probability problem involving the selection of students with specific skills from a class. We calculated the probability of selecting a Scientist, a Broadcaster, and an Artist in a particular order and then extended our analysis to consider all possible selection orders. By breaking down the problem into manageable steps and applying the principles of conditional probability and permutations, we arrived at a comprehensive solution. The final probability of selecting one student from each category, regardless of order, is rac{100}{473}. This exercise demonstrates the power of probability theory in solving real-world problems and highlights the importance of considering all possible scenarios to achieve an accurate result. The step-by-step approach not only aids in solving the problem but also enhances understanding of the underlying concepts of probability.