Ch. 4 - Probability

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 4 - Probability

Problem 4.4.28a

Chapter 4, Problem 4.4.28a

Phase I of a Clinical Trial A clinical test on humans of a new drug is normally done in three phases. Phase I is conducted with a relatively small number of healthy volunteers. For example, a phase I test of bexarotene involved only 14 subjects. Assume that we want to treat 14 healthy humans with this new drug and we have 16 suitable volunteers available.

a. If the subjects are selected and treated one at a time in sequence, how many different sequential arrangements are possible if 14 people are selected from the 16 that are available?

Verified step by step guidance

Step 1: Recognize that this is a permutation problem because the order in which the 14 subjects are selected and treated matters. In permutations, the arrangement of items is important.

Step 2: Use the formula for permutations, which is given by P(n, r) = n! / (n - r)!, where n is the total number of items (16 volunteers in this case) and r is the number of items to be arranged (14 subjects to be selected).

Step 3: Substitute the values into the formula. Here, n = 16 and r = 14, so the formula becomes P(16, 14) = 16! / (16 - 14)!.

Step 4: Simplify the denominator. Since (16 - 14)! = 2!, the formula becomes P(16, 14) = 16! / 2!.

Step 5: To compute the result, expand the factorials. For example, 16! = 16 × 15 × 14 × ... × 1, and 2! = 2 × 1. Divide 16! by 2! to find the total number of different sequential arrangements possible.

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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 counting, arrangement, and combination of objects. In this context, it helps determine how many ways we can select and arrange a subset of individuals from a larger group. Specifically, it involves understanding permutations and combinations, which are essential for solving problems related to selecting and ordering subjects in clinical trials.

Permutations

Permutations refer to the different ways in which a set of items can be arranged in order. When selecting 14 subjects from 16, the order in which they are treated matters, making this a permutation problem. The formula for permutations is n! / (n - r)!, where n is the total number of items to choose from, and r is the number of items to arrange.

Factorial

A factorial, denoted by n!, is the product of all positive integers up to n. It is a fundamental concept in combinatorics used to calculate permutations and combinations. For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. Understanding factorials is crucial for calculating the total number of arrangements when selecting and ordering subjects in a clinical trial.

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Combinations

Related Practice

Textbook Question

Corporate Officers and Committees The Self Driving Unicycle Company was recently successfully funded via Kickstarter and must now appoint a president, chief executive officer (CEO), chief operating officer (COO), and chief financial officer (CFO), and chief human resources officer (CHR). It must also appoint a strategic planning committee with five different members. There are 15 qualified candidates, and officers can also serve on the committee.

a. How many different ways can the five officers be appointed?

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Textbook Question

Redundancy in Computer Hard Drives It is generally recognized that it is wise to backup computer data. Assume that the following refer to use of Western Digital model WD60EFRX hard drives, which have an annual failure rate of 3.66% (based on data from Backblaze, Inc.).

a. If you store all of your computer data on a single hard drive, what is the probability that the drive will fail during a year?

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Textbook Question

Denomination Effect

In Exercises 13–16, use the data in the following table. In an experiment to study the effects of using four quarters versus a \$1 bill, some college students were given four quarters and others were given a \$1 bill, and they could either keep the money or spend it on gum. The results are summarized in the table (based on data from “The Denomination Effect,” by Priya Raghubir and Joydeep Srivastava, Journal of Consumer Research, Vol. 36).

Denomination Effect

a. Find the probability of randomly selecting a student who spent the money, given that the student was given four quarters.

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Textbook Question

Dice and Coins

a. Find the probability that when a single six-sided die is rolled, the outcome is 5.

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Textbook Question

Redundancy in Computer Hard Drives The Seagate ST8000NM0055 hard drive has a 1.22% rate of failures in a year (based on data from Backblaze, Inc.). For the following, assume that all hard drives are that Seagate model.

a. If all of your computer data are stored on a hard disk drive with a copy stored on a second hard disk drive, what is the probability that during a year, you can avoid catastrophe with at least one working drive? Express the result with six decimal places.

160

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Textbook Question

In Exercises 21-28, find the probability and answer the questions.

X-Linked Genetic Disease Men have XY (or YX) chromosomes and women have XX chromosomes. X-linked recessive genetic diseases (such as juvenile retinoschisis) occur when there is a defective X chromosome that occurs without a paired X chromosome that is not defective. In the following, represent a defective X chromosome with lowercase x, so a child with the xY or Yx pair of chromosomes will have the disease and a child with XX or XY or YX or xX or Xx will not have the disease. Each parent contributes one of the chromosomes to the child.

a. If a father has the defective x chromosome and the mother has good XX chromosomes, what is the probability that a son will inherit the disease?

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