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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
The total number of possible outcomes is \(8\). The sample space is \(S = \{(F, F, F), (F, F, A), (F, A, F), (F, A, A), (A, F, F), (A, F, A), (A, A, F), (A, A, A)\}\).

Step by step solution

01

Understand the options for each person

Each person can either be for or against the tax increase. Therefore, each individual can have one of two outcomes. The two outcomes for a single individual are: 'In favor' (F) and 'Against' (A).
02

Determine the total number of outcomes

Because there are two possible outcomes for each person and there are three people, a total possible outcome can be calculated as \(2^3 = 8\) outcomes.
03

List of all possible outcomes in a sample space

Here are all the possible results of the random selection of three people: \[S = \{(F, F, F), (F, F, A), (F, A, F), (F, A, A), (A, F, F), (A, F, A), (A, A, F), (A, A, A)\}\] where each tuple represents the opinion (either in favor (F) or against (A)) of three different individuals.
04

Draw a tree diagram

A tree diagram here should start with a single point (the root) that splits into two branches (for 'In favor' and 'Against'). Each branch then splits again into two (for the second person's 'In favor' and 'Against') and then those branches split another time for the third person's opinion. This way, you will end up with eight end points, each representing one of the possible outcomes mentions in the previous step.

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