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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. 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
Event A contains outcomes where books 3, 4 or 5 are picked first. Event C will contain outcomes where book 5 is picked first, or one of books 1 or 2 is picked followed by book 5.

Step by step solution

01

- Generating the Tree Diagram

All possible outcomes can be displayed in a tree diagram. Starting from the left, each branch would represent a pick. Branches would end whenever a second printing book is picked. Follow this method until all possible combinations of selections have been represented.
02

- Determine outcomes for event A

Event A requires only one book to be examined before the experiment terminates, i.e., the first book picked is a second printing. Therefore, the outcomes in Event A would be the selections where any of the second printing books (3,4,5) are picked in the first attempt.
03

- Determine outcomes for event C

Event C requires that the experiment ends when book 5, a second printing, is picked. Therefore, to get the outcomes for event C, consider each path on the tree diagram where the last book picked is book 5. Remember that the experiment stops as soon as a second printing book is chosen.

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Key Concepts

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

Random Selection
Random selection plays a crucial role in this textbook examination problem. It refers to picking books from the library with no predetermined order. The essence of random selection is its unpredictability, making it ideal for such exercises where probability trees are used. With five books available, each having an equal chance of being selected at first, the student must rely on sheer luck in choosing a second printing book, which concludes the selection process. In this example:
  • There are five books in total, each from two different printings.
  • Books 1 and 2 are first printings, while 3, 4, and 5 are second printings.
  • The task involves stopping after selecting a second printing book.
Random selection leads to the creation of a probability tree, which models possible outcomes as each book is chosen in sequence.
Event Outcome Analysis
Once the random selection process is clear, we move on to analyzing event outcomes. This is where we use a tree diagram to visualize possible sequences of book selections, particularly how and when the selection stops based on the first pick of a second printing book. Each pathway on the tree stands for a different sequence. Key points in event outcome analysis:
  • For event A, the selection ends after exactly one book, meaning a second printing book (3, 4, or 5) is picked initially.
  • The outcomes would be {3}, {4}, {5} if we consider this event.
  • In event C, the focus is on ending the sequence with the examination of book 5. This necessitates that book 5 be the first second printing book encountered, like in pathways leading to {1, 5} or {2, 5}.
Event outcome analysis allows students to see the likelihood of each scenario, supported by a clear visual reference provided by the probability tree.
Textbook Examination Problem
The textbook examination problem is a common question type in probability studies, often used to teach students the principles of probability trees and event outcomes. Here, the task revolves around visualizing outcomes and understanding conditions under which they occur, such as when a student stops examining books. In exploring the textbook examination:
  • Tree diagrams help present each possible outcome in a structured manner.
  • Individual branches show all possible selections that could happen randomly.
  • The termination condition emphasizes students鈥� understanding of probability as a real-world concept, as chances change with each pick.
Such exercises are fundamental in grasping how probability trees work, demonstrating events through a structured and methodical approach that guides students in comprehending complex random selection situations.

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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 \(P(\) A beats \(B)=.7, P(\) 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 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.)

There are two traffic lights on the route used by a certain individual to go from home to work. Let \(E\) denote the event that the individual must stop at the first light, and define the event \(F\) in a similar manner for the second light. Suppose that \(P(E)=.4, P(F)=.3\) and \(P(E \cap F)=.15\) a. What is the probability that the individual must stop at at least one light; that is, what is the probability of the event \(E \cup F ?\) b. What is the probability that the individual needn't stop at either light? c. What is the probability that the individual must stop at exactly one of the two lights? d. What is the probability that the individual must stop just at the first light? (Hint: How is the probability of this event related to \(P(E)\) and \(P(E \cap F) ?\) A Venn diagram might help.)

A medical research team wishes to evaluate two different treatments for a disease. Subjects are selected two at a time, and then one of the pair is assigned to each of the two treatments. The treatments are applied, and each is either a success (S) or a failure (F). The researchers keep track of the total number of successes for each treatment. They plan to continue the chance experiment until the number of successes for one treatment exceeds the number of successes for the other treatment by \(2 .\) For example, they might observe the results in the table below. The chance experiment would stop after the sixth pair, because Treatment 1 has 2 more successes than Treatment \(2 .\) The researchers would conclude that Treatment 1 is preferable to Treatment \(2 .\) Suppose that Treatment 1 has a success rate of \(.7\) (i.e., \(P(\) success \()=.7\) for Treatment 1 ) and that Treatment 2 has a success rate of \(.4\). Use simulation to estimate the probabilities in Parts (a) and (b). (Hint: Use a pair of random digits to simulate one pair of subjects. Let the first digit represent Treatment 1 and use \(1-7\) as an indication of a

Insurance status - covered (C) or not covered (N) \- is determined for each individual arriving for treatment at a hospital's emergency room. Consider the chance experiment in which this determination is made for two randomly selected patients. The simple events are \(O_{1}=(\mathrm{C}, \mathrm{C})\) \(O_{2}=(\mathrm{C}, \mathrm{N}), O_{3}=(\mathrm{N}, \mathrm{C})\), and \(O_{4}=(\mathrm{N}, \mathrm{N}) .\) Suppose that probabilities are \(P\left(O_{1}\right)=.81, P\left(O_{2}\right)=.09, P\left(O_{3}\right)=.09\), and \(P\left(O_{4}\right)=.01\). a. What outcomes are contained in \(A\), the event that at most one patient is covered, and what is \(P(A)\) ? b. What outcomes are contained in \(B\), the event that the two patients have the same status with respect to coverage, and what is \(P(B)\) ?

A company uses three different assembly lines \(-A_{1}\), \(A_{2}\), and \(A_{3}-\) to manufacture a particular component. Of those manufactured by \(A_{1}, 5 \%\) need rework to remedy a defect, whereas \(8 \%\) of \(A_{2}\) 's components and \(10 \%\) of \(A_{3}\) 's components need rework. Suppose that \(50 \%\) of all components are produced by \(A_{1}\), whereas \(30 \%\) are produced by \(A_{2}\) and \(20 \%\) come from \(A_{3} .\) a. Construct a tree diagram with first-generation branches corresponding to the three lines. Leading from each branch, draw one branch for rework (R) and another for no rework (N). Then enter appropriate probabilities on the branches. b. What is the probability that a randomly selected component came from \(A_{1}\) and needed rework? c. What is the probability that a randomly selected component needed rework?
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