Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 3 - Probability

문제 3.4.31

3장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.