91Ó°ÊÓ

A bookstore sells two types of books (fiction and nonfiction) in several formats (hardcover, paperback, digital, and audio). For the chance experiment that consists of observing the type and format of a single-book purchase, two possible outcomes are a hardcover fiction book and an audio nonfiction book. a. There are eight outcomes in the sample space for this experiment. List these possible outcomes. b. Do you think it is reasonable to think that the outcomes for this experiment would be equally likely? Explain. c. For customers who purchase a single book, the estimated probabilities for the different possible outcomes are given in the cells of the accompanying table. What is the probability that a randomly selected single-book purchase will be for a book in print format (hardcover or paperback)? $$ \begin{array}{l|cccc} {\text { Hardcover }} & \text { Paperback } & \text { Digital } & \text { Audio } \\ \hline \text { Fiction } & .15 & .45 & .10 & .10 \\ \text { Nonfiction } & .08 & .04 & .02 & .06 \\ \hline \end{array} $$ d. Show two different ways to compute the probability that a randomly selected single-book purchase will be for a book that is not in a print format. e. Find the probability that a randomly selected singlebook purchase will be for a work of fiction.

Step by step solution

01

Listing Possible Outcomes

The outcomes of the probability experiment are combinations of the two variables - type and format. Since there are 2 book types and 4 book formats, the count of the outcomes is \(2 \times 4 = 8\), representing all combinations of type and format.

02

Evaluating Equal Likelihood

Whether the outcomes are equally likely depends on factors like customers' preferences or store promotion, which are not given in the problem. In a random scenario, without any bias or preference, it could be assumed that all outcomes are equally likely. However, without enough context, a definite answer cannot be provided.

03

Computing Probability for Printed Format

The total probability for a purchase to be a print format (hardcover or paperback) book can be calculated by adding probabilities of all corresponding outcomes from the provided table. That is, \(P (\text{{Printed}}) = P (\text{{Fiction Hardcover}}) + P (\text{{Fiction Paperback}}) + P (\text{{Nonfiction Hardcover}}) + P (\text{{Nonfiction Paperback}}) = 0.15 + 0.45 + 0.08 + 0.04 = 0.72\).

04

Computing Probability of Non-printed Format: Method 1

The first way to calculate probability for non-printed format books (digital or audio) is similar to the previous step. Add probabilities for all respective outcomes. That is, \(P (\text{{Non-printed}}) = P (\text{{Fiction Digital}}) + P (\text{{Fiction Audio}}) + P (\text{{Nonfiction Digital}}) + P (\text{{Nonfiction Audio}}) = 0.10 + 0.10 + 0.02 + 0.06 = 0.28\).

05

Computing Probability of Non-printed Format: Method 2

The second way to calculate the probability for non-printed books is by making use of the complementary rule of probability, stating that for any event A, \(P(A') = 1 - P(A)\). Since non-printed format (digital and audio) is the complement of the printed format (hardcover and paperback), we can calculate it as \(P (\text{{Non-printed}}) = 1 - P (\text{{Printed}}) = 1 - 0.72 = 0.28\). Both methods yield the same result.

06

Compute Probability for Fiction Category

The probability that a randomly selected single-book purchase will be a work of fiction can be obtained by adding probabilities for all respective outcomes. That is, \(P (\text{{Fiction}}) = P (\text{{Fiction Hardcover}}) + P (\text{{Fiction Paperback}}) + P (\text{{Fiction Digital}}) + P (\text{{Fiction Audio}}) = 0.15 + 0.45 + 0.10 + 0.10 = 0.80\).

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Key Concepts

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

Sample Space

In probability, the sample space is the set of all possible outcomes of a random experiment. For the exercise at hand, we are dealing with a bookstore selling two types of books: fiction and nonfiction. Each book type can be purchased in one of four formats: hardcover, paperback, digital, or audio.
Hence, the sample space consists of all combinations of book types and formats, amounting to eight possible outcomes. Each outcome represents a unique combination of a type and format.

• Hardcover Fiction
• Paperback Fiction
• Digital Fiction
• Audio Fiction
• Hardcover Nonfiction
• Paperback Nonfiction
• Digital Nonfiction
• Audio Nonfiction

Listing the sample space helps in organizing and analyzing the probability of each possible outcome from the experiment.

Complementary Rule of Probability

The complementary rule of probability is a handy concept to understand. It involves the probability of the complementary event of a given event. Simply put, if you know the probability that an event will happen, you can easily figure out the probability that it won't happen.
This rule states: for any event A, the probability that A does not occur (denoted as \( P(A') \)) is equal to one minus the probability that A does occur (\( P(A) \)). Mathematically, it is expressed as: \[P(A') = 1 - P(A)\]Using this rule simplifies calculations, especially when directly calculating a probability is complex. For instance, in our bookstore exercise, the probability of selecting a non-printed format book can be found by subtracting the probability of selecting a printed format book from 1.
This concept allows for alternative approaches to determining probabilities when direct computation isn't straightforward or practical.

Equally Likely Outcomes

Equally likely outcomes refer to situations where each outcome of a random experiment has the same chance of occurring. This concept is central in classical probability, where probabilities are assigned under the assumption of complete randomness.
In the bookstore exercise, the assumption of equally likely outcomes might seem logical at first glance since there are two book types and four formats, leading to eight combinations. However, without data on customer preferences or external factors, we cannot definitively say that all outcomes are equally probable.
Real-life scenarios usually involve factors that influence likelihood, such as store promotions or popular formats. Hence, determining equal likelihood requires careful assessment of these factors, rather than assuming uniform probability across all outcomes.

Random Experiment Outcomes

A random experiment is an activity or process that produces a set of results or outcomes. In probability terms, the outcome of a random experiment is unpredictable and varies each time the experiment is conducted.
In our bookstore example, the experiment is selecting a single book with a type and format. Each outcome, such as 'Hardcover Fiction' or 'Digital Nonfiction', is an observable result of this experiment. Despite variability, randomness ensures the results are part of a well-defined sample space.
Understanding the nature of randomness is crucial when calculating probabilities. It helps in predicting likelihoods of different outcomes, assuming multiple trials over similar conditions. Probability assessments based on random experiments help anticipate results and make informed decisions.