Question

\(6.4\) A tennis shop sells five different brands of rackets, each of which comes in either a midsize version or an oversize version. Consider the chance experiment in which brand and size are noted for the next racket purchased. One possible outcome is Head midsize, and another is Prince oversize. Possible outcomes correspond to cells in the following table: $$ \begin{array}{|l|l|l|l|l|l|} \hline & \text { Head } & \text { Prince } & \text { Slazenger } & \text { Wimbledon } & \text { Wilson } \\ \hline \text { Midsize } & & & & & \\ \hline \text { Oversize } & & & & & \\ \hline \end{array} $$ a. Let \(A\) denote the event that an oversize racket is purchased. List the outcomes in \(A\). b. Let \(B\) denote the event that the name of the brand purchased begins with a W. List the outcomes in \(B\). c. List the outcomes in the event \(n o t \bar{B}\). d. Head, Prince, and Wilson are U.S. companies. Let \(C\) denote the event that the racket purchased is made by a U.S. company. List the outcomes in the event \(B\) or \(C\). e. List outcomes in \(B\) and \(C\). f. Display the possible outcomes on a tree diagram, with a first-generation branch for each brand.

Short answer

The outcomes for each event are as follows: Event A includes all oversize rackets. Event B includes all rackets from brands that start with a 'W'. Event Not B includes all rackets from brands that don't start with a 'W'. Event C includes all rackets from U.S. companies (Head, Prince, Wilson). The 'B or C' event includes all outcomes from either B or C, essentially covering all rackets from U.S. companies and Wimbledon. The 'B and C' event includes only the 'W' starting US brands.

Step-by-step solution

Define Event A

Event A is the event in which an oversize racket is purchased. For this event, the outcomes are: Head Oversize, Prince Oversize, Slazenger Oversize, Wimbledon Oversize, and Wilson Oversize.

Define Event B

Event B is the event in which the name of the brand purchased begins with a 'W'. The outcomes for this event are: Wimbledon Midsize, Wimbledon Oversize, Wilson Midsize, and Wilson Oversize.

Define Event Not B

The event ‘Not B’ refers to the event in which the name of the brand purchased does not begin with a 'W'. Thus, the outcomes for this event are the names of all remaining brands in both Midsize and Oversize: Head Midsize, Head Oversize, Prince Midsize, Prince Oversize, and Slazenger Midsize, Slazenger Oversize.

Define Event C

Event C indicates that the racket purchased is made by a U.S. company. The companies identified as being U.S. companies are Head, Prince, and Wilson. So, the outcomes for this event are: Head Midsize, Head Oversize, Prince Midsize, Prince Oversize, Wilson Midsize, and Wilson Oversize.

Define Event B or C

The event 'B or C' includes all outcomes that are either in event B or event C. Since event B's outcomes are subset of event C's outcomes, the 'B or C' event essentially includes all outcomes of event C. So, the possible outcomes of this event are the same as for event C.

Define Event B and C

The event 'B and C' would include outcomes that are common to both B and C. Outcomes here would be the 'W' starting US brands, which are Wimbledon Midsize, Wimbledon Oversize, Wilson Midsize, and Wilson Oversize.

Draw a Tree Diagram

A tree diagram can be drawn with the first generation branches representing each brand (Head, Prince, Slazenger, Wimbledon, Wilson) and the second generation branches under each brand indicating the size (Midsize, Oversize). Brand is the first decision node and Size the second, leading to 10 possible outcomes.

Key concepts

Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. It provides methods to describe and measure aspects of nature from samples. In the context of a probability experiment like the one involving the tennis racket sales, statistics comes into play by organizing the outcomes and calculating the likelihood of different events.

When evaluating chances, we encounter the probability of an event, which is the measure of how likely that event is to occur. Understanding probabilistic events in statistics is essential, as it helps anticipate the frequency of outcomes in a larger population based on a sample. For example, analyzing the sales data of tennis rackets structured by size and brand can yield insights into consumer preferences, market trends, and inventory management for the shop.

Event Outcomes

In the realm of probability experiments, an event outcome, or simply an 'outcome', is the result of a single execution of the experiment. These outcomes are the fundamental building blocks upon which probability is calculated. For instance, when a customer purchases a tennis racket, the event's outcomes are the specific combinations of brand and size of the racket purchased.

In an effort to clearly exhibit the structure of outcomes and improve student understanding, outcomes can be tabulated or illustrated graphically. For exercise improvement, it's crucial to note that listing outcomes as was done for events A, B, and C in the steps helps identify them clearly. Realizing that these outcomes can interplay in various ways to form compound events – such as 'B or C' (union of events) and 'B and C' (intersection of events) – allows further exploration into the complex nature of event probabilities.

Tree Diagram

A tree diagram is a graphical representation used to illustrate all the possible outcomes of a probability experiment step by step or stage by stage. Each branch of the tree denotes a possible decision or occurrence, which provides a path to outcomes. This makes tree diagrams extremely useful for mapping out complex probability problems where there are multiple stages or a series of events.

The tree diagram provided in the step-by-step solution shows two levels of branches: the first level for the brand and the second for the size of the racket. It's a practical way to organize and view all potential outcomes in a structured format, which is especially helpful for students who are visual learners. By following the branches, one can easily determine the total number of possible outcomes, which is essential for calculating probabilities in more complex scenarios.