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Chapter 6: Problem 7

A library has five copies of a certain textbook on reserve of which two copies ( 1 and 2 ) are first printings and the other three \((3,4\), and 5\()\) are second printings. \(\mathrm{A}\) student examines these books in random order, stopping only when a second printing has been selected. a. Display the possible outcomes in a tree diagram. b. What outcomes are contained in the event \(A\), that exactly one book is examined before the chance experiment terminates? c. What outcomes are contained in the event \(C\), that the chance experiment terminates with the examination of book 5 ?

Short Answer

Expert verified
For event \(A\), the possible outcomes are \(1-3\), \(1-4\), \(1-5\), \(2-3\), \(2-4\), and \(2-5\). For event \(C\), the possible outcomes would involve any combination of books \(1, 2, 3,\) and \(4\) (in any order), ending with book \(5\).

Step by step solution

01

Understanding Outcomes and Creating Tree Diagram

In creating the tree diagram, start with a branch for each book that the student could examine first. From each of these, draw additional branches representing the possible second books the student could examine, and so on, until all five books have been represented. The possible outcomes of the experiment are sequences in which the books might be examined. For instance, an outcome could be \(3\), or \(1-3\), or \(1-2-3\), etc., ending when the student picks up a second printing.
02

Identifying Outcomes in Event \(A\)

Event \(A\) is the event that exactly one book is examined before the experiment terminates. This means the second book picked is from the second printing. The possible outcomes for event \(A\) are therefore \(1-3\), \(1-4\), \(1-5\), \(2-3\), \(2-4\), and \(2-5\).
03

Identifying Outcomes in Event \(C\)

Event \(C\) is the event that the experiment terminates with the examination of book 5. This would mean that the student examines any combination of books \(1, 2, 3, and 4\), in any order, and finally ends with book 5. The possible outcomes include \(1-2-3-4-5\), \(1-2-4-3-5\), \(2-1-3-4-5\), up to \(5\) (where the student picks book 5 right away).

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Key Concepts

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

Tree Diagram
A tree diagram is a useful tool in probability to visualize all possible outcomes of an experiment. In this scenario, we begin by imagining each book as a branch. Each branch represents a possible choice the student could make initially. From there, we extend additional branches to represent subsequent choices.
For this exercise:
  • Begin with five initial branches, each standing for one of the books: 1, 2, 3, 4, and 5.
  • Next, from each branch associated with books 1 and 2 (the first printings), draw branches for books 3, 4, and 5, which are the second printings.
  • Continue this pattern until every book has been considered as a possible choice.
The tree diagram demonstrates sequences like 1-3 (first examine book 1, then book 3) or 3 (immediately selecting a second printing book). This visualization helps us see that the student stops examining as soon as they select a book from the second printing.
Random Selection
The concept of random selection is central to this exercise. When the student chooses a book without any preference and continues until a condition is met, that is random selection in action. Here, we assume all books have an equal chance of being picked, which adds randomness to the process.

Randomness implies the unpredictability of the order in which the student may examine the books. This is important because it influences the probability of each sequence of selections occurring. For example:
  • The chance of picking book 3 before others depends on the random turn-taking of the subsequent books.
  • If the student freely chooses a book, any book might be the first choice, which drives the need for careful event identification.
Through random selection, the learning exercise emphasizes understanding diverse combinations and their impact, showcasing the varied paths one can follow in real-life choices.
Event Identification
Event identification helps in pinpointing specific outcomes or sequences of interest within a broader set of possibilities. In this task, we explore two particular events, "Event A" and "Event C".

**Event A: One Book Examined** Event A occurs when the student stops examining books immediately after selecting the first second printing book. Possible sequences for this event include:
  • 1-3 (pick book 1, then book 3)
  • 1-4, 1-5 (similar logic with book 1 first)
  • 2-3, 2-4, 2-5 (similar logic with book 2 first)
Each outcome shows the termination just after the first second printing book is chosen. **Event C: Terminating with Book 5** Event C describes sequences where the student eventually picks book 5. Book 5 could be the first selection, or it might come after any number of other books. Possibilities include selecting all books in some order ending with book 5 like:
  • 1-2-3-4-5
  • and any combination ending with book 5
Identifying these events is vital for understanding which sequences satisfy given conditions, aiding in probability calculations.

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Most popular questions from this chapter

Define the term chance experiment, and give an example of a chance experiment with four possible outcomes.

Is ultrasound a reliable method for determining the gender of an unborn baby? The accompanying data on 1000 births are consistent with summary values that appeared in the online version of the Journal of Statistics Education ("New Approaches to Leaming Probability in the First Statistics Course" [2001]). $$ \begin{array}{ccc} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Female } \end{array} & \begin{array}{c} \text { Ultrasound } \\ \text { Predicted } \\ \text { Male } \end{array} \\ \hline \begin{array}{c} \text { Actual Gender Is } \\ \text { Female } \end{array} & 432 & 48 \\ \begin{array}{c} \text { Actual Gender Is } \\ \text { Male } \end{array} & 130 & 390 \\ \hline \end{array} $$ a. Use the given information to estimate the probability that a newborn baby is female, given that the ultrasound predicted the baby would be female. b. Use the given information to estimate the probability that a newborn baby is male, given that the ultrasound predicted the baby would be male. c. Based on your answers to Parts (a) and (b), do you think that a prediction that a baby is male and a prediction that a baby is female are equally reliable? Explain.

Consider a Venn diagram picturing two events \(A\) and \(B\) that are not disjoint. a. Shade the event \((A \cup B)^{C} .\) On a separate Venn diagram shade the event \(A^{C} \cap B^{C} .\) How are these two events related? b. Shade the event \((A \cap B)^{C} .\) On a separate Venn diagram shade the event \(A^{C} \cup B^{C} .\) How are these two events related? (Note: These two relationships together are called DeMorgan's laws.)

After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books at random to the four students \((1,2,3\), and 4\()\) who claim to have left books. One possible outcome is that 1 receives 2's book, 2 receives 4's book, 3 receives his or her own book, and 4 receives l's book. This outcome can be abbreviated \((2,4,3,1)\). a. List the 23 other possible outcomes. b. Which outcomes are contained in the event that exactly two of the books are returned to their correct owners? Assuming equally likely outcomes, what is the probability of this event? c. What is the probability that exactly one of the four students receives his or her own book? d. What is the probability that exactly three receive their own books? e. What is the probability that at least two of the four students receive their own books?

The manager of a music store has kept records of the number of CDs bought in a single transaction by customers who make a purchase at the store. The accompanying table gives six possible outcomes and the estimated probability associated with each of these outcomes for the chance experiment that consists of observing the number of CDs purchased by the next customer at the store. $$ \begin{aligned} &\begin{array}{l} \text { Number of CDs } \\ \text { purchased } \end{array} & 1 & 2 & 3 & 4 & 5 & 6 \text { or more } \\ &\begin{array}{c} \text { Estimated } \\ \text { probability } \end{array} & .45 & .25 & .10 & .10 & .07 & .03 \end{aligned} $$ a. What is the estimated probability that the next customer purchases three or fewer CDs? b. What is the estimated probability that the next customer purchases at most three CDs? How does this compare to the probability computed in Part (a)? c. What is the estimated probability that the next customer purchases five or more CDs? d. What is the estimated probability that the next customer purchases one or two CDs? e. What is the estimated probability that the next customer purchases more than two CDs? Show two different ways to compute this probability that use the probability rules of this section.
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