Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 3 - Probability

Problem 3.4.31

Chapter 3, Problem 3.4.31

31. Experiment A researcher is randomly selecting a treatment group of 10 human subjects from a group of 20 people taking part in an experiment. In how many different ways can the treatment group be selected?

Verified step by step guidance

Step 1: Recognize that this is a combination problem because the order in which the subjects are selected does not matter. The formula for combinations is given by:

C = \frac{n!}{r! (n - r)!}

, where

n

is the total number of items, and

r

is the number of items to choose.

Step 2: Identify the values of

n

and

r

. Here,

n = 20

(total number of people) and

r = 10

(number of people to select).

Step 3: Substitute the values of

n

and

r

into the combination formula:

C = \frac{20!}{10! (20 - 10)!}

Step 4: Simplify the denominator by calculating

(20 - 10)!

, which equals

10!

. The formula now becomes:

C = \frac{20!}{10! 10!}

Step 5: Cancel out the common terms in the numerator and denominator, and compute the remaining terms to find the number of ways the treatment group can be selected. This involves dividing the factorial of 20 by the product of two factorials of 10.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and permutations of objects. It provides the tools to count the number of ways to select items from a larger set without regard to the order of selection. In this context, it helps determine how many different groups of 10 can be formed from a total of 20 subjects.

Binomial Coefficient

The binomial coefficient, often denoted as 'n choose k' or C(n, k), represents the number of ways to choose k elements from a set of n elements without considering the order. It is calculated using the formula C(n, k) = n! / (k!(n-k)!), where '!' denotes factorial. This concept is essential for solving the problem of selecting the treatment group.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. In the context of this question, factorials are used in the binomial coefficient formula to compute the total number of ways to select the treatment group from the available subjects.

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Combinations

Related Practice

Textbook Question

Identifying the Sample Space of a Probability Experiment In Exercises 25-32, identify the sample space of the probability experiment and determine the number of outcomes in the sample space. Draw a tree diagram when appropriate.

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Textbook Question

The table shows the results of a survey in which 3,545,286 public and 509,168 private school teachers were asked about their full-time teaching experience.

Are the events “being a public school teacher” and “having more than 20 years of full-time teaching experience” independent? Explain.

Textbook Question

In Exercises 7-14, perform the indicated calculation.

9.8C3

Textbook Question

In Exercises 15-18, determine whether the situation involves permutations, combinations, or neither. Explain your reasoning.

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Textbook Question

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Textbook Question

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A person who earned a degree in the year is randomly selected. Find the probability that the degree earned by the person is a

a. bachelor's degree.