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Chapter 6: Problem 15

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.

Short Answer

Expert verified
a. Possible outcomes are: Hardcover Fiction, Paperback Fiction, Digital Fiction, Audio Fiction, Hardcover Nonfiction, Paperback Nonfiction, Digital Nonfiction, Audio Nonfiction. b. Equal likelihood of outcomes depends on customers' preferences or store promotion which we don't have information on. c. The probability for a book purchase to be a printed format is 0.72. d. The two different methods both yield a probability of 0.28 for a book not in a print format. e. The probability for a purchase to be a work of fiction is 0.80.

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.

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Most popular questions from this chapter

Three friends \((A, B\), and \(C)\) will participate in a round-robin tournament in which each one plays both of the others. Suppose that \(P(\mathrm{~A}\) beats \(\mathrm{B})=.7, P(\mathrm{~A}\) beats C) \(=.8, P(\mathrm{~B}\) beats \(\mathrm{C})=.6\), and that the outcomes of the three matches are independent of one another. a. What is the probability that \(\mathrm{A}\) wins both her matches and that \(\mathrm{B}\) beats \(\mathrm{C}\) ? b. What is the probability that \(A\) wins both her matches? c. What is the probability that \(A\) loses both her matches? d. What is the probability that each person wins one match? (Hint: There are two different ways for this to happen.)

The student council for a school of science and math has one representative from each of the five academic departments: biology (B), chemistry (C), mathematics (M), physics (P), and statistics (S). Two of these students are to be randomly selected for inclusion on a university-wide student committee (by placing five slips of paper in a bowl, mixing, and drawing out two of them). a. What are the 10 possible outcomes (simple events)? b. From the description of the selection process, all outcomes are equally likely. What is the probability of each simple event? c. What is the probability that one of the committee members is the statistics department representative? d. What is the probability that both committee members come from laboratory science departments?

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The report "Twitter in Higher Education: Usage Habits and Trends of Today's College Faculty" (Magna Publications, September 2009) describes results of a survey of nearly 2000 college faculty. The report indicates the following: \- \(30.7 \%\) reported that they use Twitter and \(69.3 \%\) said that they did not use Twitter. \- Of those who use Twitter, \(39.9 \%\) said they sometimes use Twitter to communicate with students. \- Of those who use Twitter, \(27.5 \%\) said that they sometimes use Twitter as a learning tool in the classroom. Consider the chance experiment that selects one of the study participants at random and define the following events: \(T=\) event that selected faculty member uses Twitter \(C=\) event that selected faculty member sometimes uses Twitter to communicate with students \(L=\) event that selected faculty member sometimes uses Twitter as a learning tool in the classroom a. Use the given information to determine the following probabilities: i. \(\quad P(T)\) ii. \(P\left(T^{C}\right)\) iii. \(P(C \mid T)\) iv. \(P(L \mid T)\) v. \(P(C \cap T)\) b. Interpret each of the probabilities computed in Part (a). c. What proportion of the faculty surveyed sometimes use Twitter to communicate with students? [Hint: Use the law of total probability to find \(P(C) .]\) d. What proportion of faculty surveyed sometimes use Twitter as a learning tool in the classroom?

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