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Chapter 4: Problem 8

In a group of people, some are in favor of a tax increase on rich people to reduce the federal deficit and others are against it. (Assume that there is no other outcome such as "no opinion" and "do not know.") Three persons are selected at random from this group and their opinions in favor or against raising such taxes are noted. How many total outcomes are possible? Write these outcomes in a sample space \(S\). Draw a tree diagram for this experiment.

Short Answer

Expert verified
There are 8 possible outcomes of this experiment, represented by the following sample space \(S\): {(F, F, F), (F, F, A), (F, A, F), (F, A, A), (A, F, F), (A, F, A), (A, A, F), and (A, A, A)}. A tree diagram for this experiment would have 8 end points, each representing one of the outcomes in the sample space.

Step by step solution

01

Determine the Possible Outcomes for Each Person

Each person selected can either be in favor of the tax increase (F) or against it (A). Thus, there are two possible outcomes for each person's response.
02

Define the Sample Space

The sample space \(S\) is the set of all possible outcomes of this experiment. Since we are selecting three people, the possible outcomes become combinations of the responses of these three people. These are: (F, F, F), (F, F, A), (F, A, F), (F, A, A), (A, F, F), (A, F, A), (A, A, F), and (A, A, A).
03

Draw a Tree Diagram

To represent these outcomes visually, we can draw a tree diagram. Starting from a single point, draw two branches (representing Person 1's responses) labelled F and A, each ending in two further branches (representing Person 2's responses). Continue this process one more time for Person 3's responses. The eight end points of the tree will represent the eight possible outcomes in the sample space.

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Key Concepts

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

Sample Space
In probability theory, a sample space is a comprehensive set that includes all possible outcomes of a statistical experiment. In the context of our problem, we are examining the opinions of three people on a tax increase. Each person can either be in favor (F) or against (A) the measure. By combining these individual choices, the sample space is created.
  • The total outcomes for this scenario are represented as tuples, such as (F, F, F) and (A, A, A).
  • These tuples represent every outcome where each position corresponds to one person's opinion.
  • In essence, for three people, each having two possible outcomes, there are a total of 2 x 2 x 2 = 8 possible combinations.
Understanding the sample space is crucial, as it lays the foundation for calculating the probability of any specific event occurring within this space.
Combinatorics
Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination possibilities. In our exercise, combinatorics helps us systematically count all potential outcomes when people express their opinions.
  • With three people and two options (F or A) for each person, we determine the number of combinations using the formula for total outcomes: \(2^n\), where \(n\) is the number of people involved.
  • For our scenario, this results in \(2^3 = 8\) different combinations.
  • Combinatorics provides the tools needed to understand how to organize and evaluate these possible outcomes.
Mastering combinatorics enables you to tackle more complex probability problems in a structured way, often revealing insightful patterns along the way.
Binomial Outcomes
A binomial outcome is a specific kind of result that can occur in only one of two ways: a success or a failure. In our scenario, being in favor (F) can be considered one outcome, while against (A) can be considered the other. Since each person has these two distinct outcomes, this situation is perfectly suited for a binomial probability model.
  • Think of each person's opinion as a trial that results in one of two possible outcomes.
  • If you need to find the probability of any specific sequence of responses, you apply the binomial probability formula.
With binomial outcomes, predictions and probabilities can be calculated by considering each trial's independent probability. This can include scenarios beyond just opinions, making the concept widely applicable.
Tree Diagram
A tree diagram is a visual representation used to map out all possible outcomes of a sequence of events. It is especially helpful in probability theory to illustrate and simplify the process of calculating outcomes.
  • In our problem, the tree diagram begins with an initial point, leading to the two possibilities for the first person's opinion: F or A.
  • Each branch then splits again to represent the two possible outcomes for the second person, and again for the third person.
  • Ultimately, this results in eight unique end points, each correlating with a different outcome in the sample space.
Creating a tree diagram allows you to visually trace each path, making it easier to comprehend how the sample space is constructed and how probabilities are derived from it. Such diagrams act as a powerful tool not only for simple exercises but for complex scenarios as well.

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Most popular questions from this chapter

Two thousand randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives a two-way classification of the responses. $$ \begin{array}{lcc} \hline & \text { Have Shopped } & \text { Have Never Shopped } \\ \hline \text { Male } & 500 & 700 \\ \text { Female } & 300 & 500 \\ \hline \end{array} $$ a. If one adult is selected at random from these 2000 adults, find the probability that this adult i. has never shopped on the Internet ii. is a male iii. has shopped on the Internet given that this adult is a female iv. is a male given that this adult has never shopped on the Internet b. Are the events "male" and "female" mutually exclusive? What about the events "have shopped" and "male?" Why or why not? c. Are the events "female" and "have shopped" independent? Why or why not?

The following table gives a two-way classification of all basketball players at a state university who began their college careers between 2004 and 2008 , based on gender and whether or not they graduated. $$ \begin{array}{lcc} \hline & \text { Graduated } & \text { Did Not Graduate } \\ \hline \text { Male } & 126 & 55 \\ \text { Female } & 133 & 32 \\ \hline \end{array} $$ If one of these players is selected at random, find the following probabilities. a. \(P\) (female or did not graduate) b. \(P(\) graduated or male \()\)

In a statistics class of 42 students, 28 have volunteered for community service in the past. If two students are selected at random from this class, what is the probability that both of them have volunteered for community service in the past? Draw a tree diagram for this problem.

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is \(.13 .\) If the patient does not have the disease, the probability that the test indicates a (false) positive is .10. Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive (Hint: A tree diagram may be helpful in part c.)

Draw a tree diagram for three tosses of a coin. List all outcomes for this experiment in a sample space \(S\)
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