Question

A college library has four copies of a certain book; the copies are numbered \(1,2,3\), and 4 . Two of these are selected at random. The first selected book is placed on 2 -hr reserve, and the second book can be checked out overnight. a. Construct a tree diagram to display the 12 outcomes in the sample space. b. Let \(A\) denote the event that at least one of the books selected is an even-numbered copy. What outcomes are in \(A ?\) c. Suppose that copies 1 and 2 are first printings, whereas copies 3 and 4 are second printings. Let \(B\) denote the event that exactly one of the copies selected is a first printing. What outcomes are contained in \(B\) ?

Short answer

There are 12 possible outcomes when selecting the books. For event \(A\), the outcomes are: (1,2), (1,4), (2,1), (2,3), (2,4), (3,2), (3,4), (4,1), (4,2), (4,3). For event \(B\) the outcomes are: (1,3), (1,4), (2,3), (2,4), (3,1), (3,2), (4,1), (4,2).

Step-by-step solution

Create Tree Diagram

To understand the different possible outcomes it's first necessary to construct a tree diagram. In this case, there are four books numbered from 1 to 4, and two books are selected in succession. Thus, for the first selection there are 4 options and for each of these, there are 3 different books that can be selected second, giving a total of \(4 \times 3 = 12\) paths or outcomes.

Identifying Outcomes in Event \(A\)

Event \(A\) means that at least one out of the two books selected has an even number; i.e. book number 2 or 4. The outcomes that meet these conditions are: (1,2), (1,4), (2,1), (2,3), (2,4), (3,2), (3,4), (4,1), (4,2), (4,3)

Identifying Outcomes in Event \(B\)

Event \(B\) is the event that exactly one of the books selected is a first printing. In this case books numbered 1 and 2 are first printings and books numbered 3 and 4 are second printings. The outcomes for this event where exactly one book is first printing are: (1,3), (1,4), (2,3), (2,4), (3,1), (3,2), (4,1), (4,2)

Key concepts

Sample Space in Probability

In probability, the sample space is defined as the set of all possible outcomes of a random experiment. When considering the selection of books from the college library example, the sample space consists of all the possible pairs of books that could be selected.

To visualize this, picture each book as a potential choice in a sequence. With four books, labeled 1 through 4, the first choice has four possibilities. After a book is selected for the reserve, the second choice will then have three possibilities, since one of the books is no longer available. This leads to the calculation of the total number of outcomes as the product of the possibilities for each step: 4 choices initially multiplied by 3 choices subsequently gives us 12 possible outcomes in the sample space.

The importance of defining the sample space lies in its role as the foundation of determining probabilities. Any event we consider is a subset of this sample space, and the likelihood of any given event is found by comparing the number of favorable outcomes to the total number of outcomes in the sample space.

Event Outcomes in Statistics

Event outcomes in statistics refer to the results that satisfy the conditions of a specific event within an experiment. In the context of the book selection, an event might be defined as selecting at least one even-numbered copy, which was represented as event A.

To identify the outcomes that belong to event A, we look for all pairs that include either the number 2 or number 4. It's important to note that order matters here since the first book is placed on 2-hour reserve and the second is available for overnight checkout. This results in ten pairs where at least one of the books is an even number, indicating a high probability of event A occurring.

Understanding event outcomes allows us to calculate the probability of an event by comparing the number of event outcomes to the total number of outcomes in the sample space. It also helps to clarify the nature of an event — whether it's dependent on order, the presence of a particular element, or exclusivity of conditions, for example.

First and Second Printings Probability

In our situation, considering the different printings of the books introduces a new event, event B, which concerns the selection of exactly one first printing copy. This event encapsulates a more specific condition: among the two books, one must be a first printing (copy 1 or 2), and the other must be a second printing (copy 3 or 4).

Here, we've identified that there are eight outcomes that fit this event, as indicated in the step-by-step solution. This event highlights a unique probability situation where we are dealing with a combination of criteria that restrict our event outcomes to specific pairings.

By calculating the probability of selecting exactly one first printing, we also indirectly learn about the likelihood of both or neither of the books being a first printing. This is valuable in statistics as it allows for a deeper understanding of how different events relate to each other within the same sample space and how specific conditions can drastically alter probabilities.