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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

Refer to the following information on births in the United States over a given period of time: $$ \begin{array}{lr} \text { Type of Birth } & \text { Number of Births } \\ \hline \text { Single birth } & 41,500,000 \\ \text { Twins } & 500,000 \\ \text { Triplets } & 5,000 \\ \text { Quadruplets } & 100 \\ \hline \end{array} $$ Use this information to approximate the probability that a randomly selected pregnant woman who reaches full term a. Delivers twins b. Delivers quadruplets c. Gives birth to more than a single child

Jeanie is a bit forgetful, and if she doesn't make a "to do" list, the probability that she forgets something she is supposed to do is .1. Tomorrow she intends to run three errands, and she fails to write them on her list. a. What is the probability that Jeanie forgets all three errands? What assumptions did you make to calculate this probability? b. What is the probability that Jeanie remembers at least one of the three errands? c. What is the probability that Jeanie remembers the first errand but not the second or third?

The article "Chances Are You Know Someone with a Tattoo, and He's Not a Sailor" (Associated Press, June 11, 2006) included results from a survey of adults aged 18 to 50 . The accompanying data are consistent with summary values given in the article. $$ \begin{array}{l|cc} & \text { At Least One Tattoo } & \text { No Tattoo } \\ \hline \text { Age 18-29 } & 18 & 32 \\ \text { Age 30-50 } & 6 & 44 \\ \hline \end{array} $$ Assuming these data are representative of adult Americans and that an adult American is selected at random, use the given information to estimate the following probabilities. a. \(P(\) tattoo \()\) b. \(P(\) tattoo \(\mid\) age \(18-29)\) c. \(P(\) tattoo \(\mid\) age \(30-50\) ) d. \(P(\) age \(18-29 \mid\) tattoo \()\)

Consider the chance experiment in which both tennis racket head size and grip size are noted for a randomly selected customer at a particular store. The six possible outcomes (simple events) and their probabilities are displayed in the following table: $$ \begin{array}{llll} && {\text { Grip Size }} \\ \text { Head size } & 4_{8}^{\frac{3}{8}} \text { in. } & 4 \frac{1}{2} \text { in. } & 4 \frac{5}{8} \text { in. } \\ \hline \text { Midsize } & O_{1}(.10) & O_{2}(.20) & O_{3}(.15) \\ \text { Oversize } & O_{4}(.20) & O_{5}(.15) & O_{6}(.20) \\ \hline \end{array} $$ a. The probability that grip size is \(4 \frac{1}{2}\) inches (event \(A\) ) is \(P(A)=P\left(O_{2}\right.\) or \(\left.O_{5}\right)=.20+.15=.35\). How would you interpret this probability? b. Use the result of Part (a) to calculate the probability that grip size is not \(4 \frac{1}{2}\) inches c. What is the probability that the racket purchased has an oversize head (event \(B\) ), and how would you interpret this probability? d. What is the probability that grip size is at least \(4 \frac{1}{2}\) inches?

The National Public Radio show Car Talk has a feature called "The Puzzler." Listeners are asked to send in answers to some puzzling questions-usually about cars but sometimes about probability (which, of course, must account for the incredible popularity of the program!). Suppose that for a car question, 800 answers are submitted, of which 50 are correct. a. Suppose that the hosts randomly select two answers from those submitted with replacement. Calculate the probability that both selected answers are correct. (For purposes of this problem, keep at least five digits to the right of the decimal.) b. Suppose now that the hosts select the answers at random but without replacement. Use conditional probability to evaluate the probability that both answers selected are correct. How does this probability compare to the one computed in Part (a)?
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