Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo

Larson 8th Edition

Ch. 3 - Probability

Problema 3.4.31

Capitolo 3, Problema 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?

Guida verificata passo dopo passo

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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Concetti chiave

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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.

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Percorso guidato

06:14

Coefficient of Determination

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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Percorso guidato

05:22

Combinations

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