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Chapter 4: Problem 1

An experiment has three steps with three outcomes possible for the first step, two outcomes possible for the second step, and four outcomes possible for the third step. How many experimental outcomes exist for the entire experiment?

Short Answer

Expert verified
There are 24 experimental outcomes for the entire experiment.

Step by step solution

01

Calculate Outcomes for Each Step

First, we determine the number of possible outcomes in each step of the experiment. For the first step, there are 3 outcomes; for the second step, there are 2 outcomes; and for the third step, there are 4 outcomes.
02

Multiply Outcomes

To find the total number of experimental outcomes for all three steps combined, you multiply the number of outcomes from each step together. This is because each combination of outcomes from the steps forms a unique experimental outcome.
03

Compute Total Outcomes

Perform the multiplication: 3 outcomes from the first step multiplied by 2 outcomes from the second step multiplied by 4 outcomes from the third step, i.e., \(3 \times 2 \times 4\).
04

Simplify the Expression

Calculate the result: \(3 \times 2 = 6\), then multiply the result by 4, \(6 \times 4 = 24\).
05

Interpret the Result

The total number of experimental outcomes for the entire experiment is 24. This means there are 24 possible combinations of outcomes for the experiment as a whole.

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Key Concepts

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

Experimental Outcomes
Experimental outcomes are the different results that can be observed or measured as a part of an experiment. Each possible result of an experiment is an outcome. When you conduct an experiment with several steps, each step contributes its own set of potential outcomes.
For example, in our experiment, the first step offers 3 possible outcomes, the second step provides 2, and the third step presents 4 different outcomes. Each step's outcomes are independent of the others. Together, they define the entire range of what can happen during the course of the experiment.
Understanding experimental outcomes is crucial because it allows us to anticipate the scope of an experiment. This anticipation helps to plan, analyze, and interpret the various scenarios that might occur during the experimentation.
Steps in Experiments
Understanding the distinct steps in an experiment is key to identifying all possible outcomes. Each step in an experiment can have its set of possible outcomes.
For instance, consider an experiment with three steps: the first step might be like rolling a die (three outcomes), selecting a color from a bag (two outcomes), and finally flipping a coin (four possible outcomes).
To systematically map out an experiment, break down the following:
  • What happens or is measured in each step?
  • How do these steps interact?
  • How many outcomes can occur in each step independently?
Each decision point or random event can change the course of the experiment. By structuring the experiment into its logical steps, you can better manage and understand the multitude of possible results.
Multiplication Principle in Statistics
The multiplication principle is a fundamental concept in combinatorics. It tells us how to calculate the total number of possible outcomes for a series of independent events, like steps in our experiment.
If you know the number of possible outcomes for each independent step, you can use the multiplication principle to find the total number of possible outcomes for all steps combined. Simply multiply the number of outcomes from each step together.
For our experiment, there are three steps:
  • The first step has 3 outcomes.
  • The second step offers 2 outcomes.
  • The third step provides 4 outcomes.
Using the multiplication principle, we calculate the total number of outcomes by multiplying: \(3 \times 2 \times 4\). This operation results in 24 possible experimental outcomes.
This principle is incredibly useful in probability and statistics, as it allows the examination of complex experiments by breaking them down into manageable components.

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Most popular questions from this chapter

Suppose that we have two events, \(A\) and \(B,\) with \(P(A)=.50, P(B)=.60,\) and \(P(A \cap B)=.40\) a. Find \(P(A | B)\) b. Find \(P(B | A)\) c. Are \(A\) and \(B\) independent? Why or why not?

Simple random sampling uses a sample of size \(n\) from a population of size \(N\) to obtain data that can be used to make inferences about the characteristics of a population. Suppose that, from a population of 50 bank accounts, we want to take a random sample of four accounts in order to learn about the population. How many different random samples of four accounts are possible?

The U.S. population by age is as follows (The World Almanac, 2004 ). The data are in millions of people. $$\begin{array}{lc} \text { Age } & \text { Number } \\ 19 \text { and under } & 80.5 \\ 20 \text { to } 24 & 19.0 \\ 25 \text { to } 34 & 39.9 \\ 35 \text { to } 44 & 45.2 \\ 45 \text { to } 54 & 37.7 \\ 55 \text { to } 64 & 24.3 \\ 65 \text { and over } & 35.0 \end{array}$$ Assume that a person will be randomly chosen from this population. a. What is the probability the person is 20 to 24 years old? b. What is the probability the person is 20 to 34 years old? c. What is the probability the person is 45 years or older?

An experiment with three outcomes has been repeated 50 times, and it was learned that \(E_{1}\) occurred 20 times, \(E_{2}\) occurred 13 times, and \(E_{3}\) occurred 17 times. Assign probabilities to the outcomes. What method did you use?

In early \(2003,\) President Bush proposed eliminating the taxation of dividends to shareholders on the grounds that it was double taxation. Corporations pay taxes on the earnings that are later paid out in dividends. In a poll of 671 Americans, TechnoMetrica Market Intelligence found that \(47 \%\) favored the proposal, \(44 \%\) opposed it, and \(9 \%\) were not sure (Investor's Business Daily, January 13,2003 ). In looking at the responses across party lines the poll showed that \(29 \%\) of Democrats were in favor, \(64 \%\) of Republicans were in favor, and \(48 \%\) of Independents were in favor. a. How many of those polled favored elimination of the tax on dividends? b. What is the conditional probability in favor of the proposal given the person polled is a Democrat? c. Is party affiliation independent of whether one is in favor of the proposal? d. If we assume people's responses were consistent with their own self- interest, which group do you believe will benefit most from passage of the proposal?
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