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Chapter 7: Problem 8

List the simple events associated with each experiment. A meteorologist preparing a weather map classifies the expected average temperature in each of five neighboring states (MN, WI, IA, IL, MO) for the upcoming week as follows: a. More than \(10^{\circ}\) below average b. Normal to \(10^{\circ}\) below average c. Higher than normal to \(10^{\circ}\) above average d. More than \(10^{\circ}\) above average Using each state's abbreviation and the categories-(a),(b), (c), and (d) - the meteorologist records these data.

Short Answer

Expert verified
The 20 simple events associated with the experiment are: (MN,a), (MN,b), (MN,c), (MN,d), (WI,a), (WI,b), (WI,c), (WI,d), (IA,a), (IA,b), (IA,c), (IA,d), (IL,a), (IL,b), (IL,c), (IL,d), (MO,a), (MO,b), (MO,c), and (MO,d).

Step by step solution

01

Identify the factors

There are two factors in the experiment: states and temperature categories. There are 5 states (MN, WI, IA, IL, MO) and 4 temperature categories (a, b, c, d).
02

Calculate the total number of possible outcomes

To find out the total number of possible outcomes, we can use the multiplication principle, which is multiplying the number of possibilities for each factor. In this case, there are 5 states with 4 possible classifications each. Therefore, the total number of possible outcomes is 5 states * 4 categories = 20 outcomes.
03

List all possible outcomes

Now that we know there are 20 possible outcomes, we'll list each possible combination. We will represent each simple event in the experiment as (state abbreviation, temperature category), such as (MN,a) or (WI,c). 1. (MN,a) 2. (MN,b) 3. (MN,c) 4. (MN,d) 5. (WI,a) 6. (WI,b) 7. (WI,c) 8. (WI,d) 9. (IA,a) 10. (IA,b) 11. (IA,c) 12. (IA,d) 13. (IL,a) 14. (IL,b) 15. (IL,c) 16. (IL,d) 17. (MO,a) 18. (MO,b) 19. (MO,c) 20. (MO,d) These are the 20 simple events associated with the experiment.

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Key Concepts

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

Probability Theory
Probability theory is a fundamental aspect of mathematics that deals with the analysis of random events. The likelihood that a particular event will occur is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

Understanding probability helps to predict the likelihood of outcomes and can guide decision-making in uncertain situations. For example, a meteorologist uses probability to forecast weather conditions by classifying expected average temperatures and assessing the chances for various scenarios. Similarly, when we list the simple events for their predictions, we're adopting the foundations of probability theory to represent all potential outcomes in a structured way.
Multiplication Principle
The multiplication principle, also known as the fundamental counting principle, is a key concept in probability that allows us to determine the total number of possible outcomes in an experiment when there are multiple stages or factors. According to this principle, if one event can occur in 'm' ways and a subsequent event can occur in 'n' ways, the total number of ways the two events can occur is given by the product 'm x n'.

In the given exercise, this principle is applied by multiplying the number of states (5) by the number of temperature categories (4), resulting in a total of 20 distinct outcomes. Using this systematic approach simplifies the process of identifying all possible combinations without the need to list them out individually, ensuring no possibilities are overlooked.
Sample Space
In probability, the sample space is the complete set of all possible outcomes of a random experiment. It's an essential building block for determining probabilities, as any event is a subset of this space. When we define the sample space, we can better visualize the structure of an experiment and conduct probability analysis with more precision.

For instance, the exercise mentioned earlier requires listing the simple events for temperature forecasting across five states. The sample space consists of every possible (state, temperature category) pair, and here, it contains 20 elements. These elements represent the foundation upon which probability calculations are made, particularly when identifying the likelihood of simple events within a more complex collection of possibilities.

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Most popular questions from this chapter

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