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Chapter 2: Problem 6

A college library has five copies of a certain text on reserve. 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. One possible outcome is 5 , and another is 213 . a. List the outcomes in \(\mathcal{}\) b. Let \(A\) denote the event that exactly one book must be examined. What outcomes are in \(A\) ? c. Let \(B\) be the event that book 5 is the one selected. What outcomes are in \(B\) ? d. Let \(C\) be the event that book 1 is not examined. What outcomes are in \(C\) ?

Short Answer

Expert verified
a. 3, 4, 5, 13, 23, 14, 24, 15, 25 b. 3, 4, 5 c. 5, 15, 25 d. 3, 4, 5, 23, 24, 25

Step by step solution

01

Identify All Possible Outcomes

The student stops examining books as soon as a second printing is selected. The second printings are book 3, book 4, and book 5. For each of these books, the student can select them as the first book, or after one of books 1 or 2, which are first printings. Therefore, the possible outcomes are: - 3 - 4 - 5 - 13 - 23 - 14 - 24 - 15 - 25.
02

Determine Outcomes for Event A

Event A is the event that exactly one book must be examined. This means the first book examined must be a second printing (book 3, book 4, or book 5). Therefore, the outcomes in event A are: - 3 - 4 - 5.
03

Determine Outcomes for Event B

Event B is the event that book 5 is the one selected. This can happen either when book 5 is selected first or after one of the first printings. Therefore, the outcomes in event B are: - 5 - 15 - 25.
04

Determine Outcomes for Event C

Event C is the event that book 1 is not examined. This means that book 1 is not in any of the selections before a second printing. Therefore, the outcomes in event C are: - 3 - 4 - 5 - 23 - 24 - 25.

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Key Concepts

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

Random Order Selection
Random order selection is a concept where items are chosen without any specific sequence. In the library example, the student examines books in no predetermined order. This means that, as each book is considered, each possible choice is equally likely.

Understanding how random order affects outcomes is crucial in probability. For instance, the student could start with any of the five books. Different combinations occur purely by chance. In this context, the randomness forms the basis for calculating probabilities and determining possible outcomes.
Event Outcomes
Event outcomes refer to the different possibilities that can occur when an event takes place. In our library scenario, an event outcome is the sequence in which the student selects the books.

To solve such problems, it's important to list all potential outcomes. Each sequence where the student stops at a second printing is an outcome. For instance, in this case, some possible outcomes include because the stopping point is determined by the first occurrence of a second printing. Each unique sequence where the condition is met counts as a separate outcome.
Probability Events
Probability events are specific scenarios that we are interested in when evaluating the likelihood of a certain outcome. In our example, defining events like Event A, B, and C helps in understanding which outcomes belong to which scenarios.

For Example:
  • Event A: The event that exactly one book is examined.
  • Event B: The event that book 5 is selected.
  • Event C: The event that book 1 is not examined.
By categorizing outcomes into events, it becomes simpler to calculate and understand what the probability of each event occurring is. This grouping aids in simplifying the analysis of complex situations.
Permutations in Probability
Permutations in probability involve arranging a set of items in different sequences. This concept is particularly relevant when the order of selection matters. It allows for the consideration of all possible sequences of outcomes.

In the library example, permutations help articulate the different ways books can be examined, especially when multiple first and second printings are involved. Each distinct permutation is a potential outcome with its unique probability. When one book is chosen, it alters the likelihood of subsequent choices, making permutations a key consideration in calculating correct probabilities.

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Most popular questions from this chapter

For any events \(A\) and \(B\) with \(P(B)>0\), show that \(P(A \mid B)+P\left(A^{\prime} \mid B\right)=1\).

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