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Chapter 5: Problem 61

The grade appeal process at a university requires that a jury be structured by selecting five individuals randomly from a pool of eight students and ten faculty. (a) What is the probability of selecting a jury of all students? (b) What is the probability of selecting a jury of all faculty? (c) What is the probability of selecting a jury of two students and three faculty?

Short Answer

Expert verified
0.0065, 0.0294, and 0.3922

Step by step solution

01

- Determine Total Number of People

There are 8 students and 10 faculty in the pool. Therefore, the total number of people is 8 + 10 = 18.
02

- Calculate Total Possible Juries

The number of ways to choose 5 individuals out of 18 is given by the combination formula \[ C(n, k) = \frac{n!}{k! (n - k)!} \] where \( n \) is 18 and \( k \) is 5. Calculating this gives \[ C(18, 5) = \frac{18!}{5! \times 13!} = 8568 \].
03

- Compute Probability of Selecting All Students

The number of ways to choose 5 students out of 8 is given by \[ C(8, 5) = \frac{8!}{5! \times 3!} = 56 \]. The probability is then \[ P(\text{all students}) = \frac{C(8, 5)}{C(18, 5)} = \frac{56}{8568} \approx 0.0065. \]
04

- Compute Probability of Selecting All Faculty

The number of ways to choose 5 faculty out of 10 is given by \[ C(10, 5) = \frac{10!}{5! \times 5!} = 252 \]. The probability is then \[ P(\text{all faculty}) = \frac{C(10, 5)}{C(18, 5)} = \frac{252}{8568} \approx 0.0294. \]
05

- Compute Probability of Selecting 2 Students and 3 Faculty

The number of ways to choose 2 students out of 8 is \[ C(8, 2) = \frac{8!}{2! \times 6!} = 28 \]. The number of ways to choose 3 faculty out of 10 is \[ C(10, 3) = \frac{10!}{3! \times 7!} = 120 \]. The probability is then \[ P(\text{2 students, 3 faculty}) = \frac{C(8, 2) \times C(10, 3)}{C(18, 5)} = \frac{28 \times 120}{8568} \approx 0.3922. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

combinations
Combinations are a fundamental concept in probability and statistics. They allow us to determine how many different ways we can choose a specific number of items from a larger set.
For example, in our problem, we're selecting 5 jurors from a pool of 18 people. We use the combination formula \( C(n, k) = \frac{n!}{k! (n - k)!} \) where n is the total number of items, and k is the number to choose.
This formula helps us calculate the number of possible groups without regard to the order in which they are chosen.
Using this formula, we found that there are 8568 possible combinations for selecting the jury.
probability theory
Probability theory helps us quantify the likelihood of an event occurring.
It's expressed as a number between 0 and 1, where 0 means the event cannot happen, and 1 means it will definitely happen.
In our exercise, we used probability theory to calculate the chances of different jury compositions.
For instance, the probability of selecting a jury of all students is given by the fraction of favorable combinations over the total combinations. Specifically, this is calculated as:
\( P(\text{all students}) = \frac{C(8, 5)}{C(18, 5)} = \frac{56}{8568} \)
Probability theory helps us understand and predict outcomes based on known data and assumptions.
statistical methods
Statistical methods involve collecting, analyzing, interpreting, presenting, and organizing data.
In this problem, we used several statistical techniques to solve for the probabilities.
First, we counted the total individuals and possible juries using combinations.
Then, for each specific group makeup (all students, all faculty, two students and three faculty), we calculated the corresponding probabilities.
These methods help us make informed decisions and predictions.
If you understand how to use these steps, you can apply them to a wide range of problems.
This makes statistical methods very powerful in research and daily decision-making.
higher education statistics
Higher education statistics involve more advanced concepts and applications.
In this context, students must grasp foundational statistics, probability theory, and their practical applications.
The exercise exemplifies real-world scenarios where such skills are crucial.
For instance, universities need to make informed decisions about jury selections, admissions, and resource allocation.
Understanding and utilizing these principles ensures fairness and accuracy.
It also enhances critical thinking and analytical skills, which are essential for academic and professional success.
Practical exercises like these are invaluable for mastering complex concepts in higher education.

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