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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

According to a survey of 2000 home owners, 800 of them own homes with three bedrooms, and 600 of them own homes with four bedrooms. If one home owner is selected at random from these 2000 home owners, find the probability that this home owner owns a house that has three or four bedrooms. Explain why this probability is not equal to \(1.0\)

A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to \(3599 .\) To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

Jason and Lisa are planning an outdoor reception following their wedding. They estimate that the probability of bad weather is \(.25\), that of a disruptive incident (a fight breaks out, the limousine is late, etc.) is \(.15\), and that bad weather and a disruptive incident will occur is. 08 . Assuming these estimates are correct, find the probability that their reception will suffer bad weather or a disruptive incident.

Refer to Exercise 4.52, which contains information on a July 21, 2009 www.HuffingtonPost.com survey that asked people to choose their favorite junk food from a list of choices. The following table contains results classified by gender. (Note: There are 4801 females and 3201 males.) $$ \begin{array}{lcc} \hline \text { Favorite Junk Food } & \text { Female } & \text { Male } \\ \hline \text { Chocolate } & 1518 & 531 \\ \text { Sugary candy } & 218 & 127 \\ \text { Ice cream } & 685 & 586 \\ \text { Fast food } & 312 & 463 \\ \text { Cookies } & 431 & 219 \\ \text { Chips } & 458 & 649 \\ \text { Cake } & 387 & 103 \\ \text { Pizza } & 792 & 523 \\ \hline \end{array} $$ a. Suppose that one person is selected at random from this sample of 8002 respondents. Find the following probabilities. i. Probability of the intersection of events female and ice cream. ii. Probability of the intersection of events male and pizza. b. Mention at least four other joint probabilities you can calculate for this table and then find their probabilities. You may draw a tree diagram to find these probabilities.

An appliance repair company that makes service calls to customers' homes has found that \(5 \%\) of the time there is nothing wrong with the appliance and the problem is due to customer error (appliance unplugged, controls improperly set, etc.). Two service calls are selected at random, and it is observed whether or not the problem is due to customer error. Draw a tree diagram. Find the probability that in this sample of two service calls a. both problems are due to customer error b. at least one problem is not due to customer error
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