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Chapter 15: Problem 61

Explain the concepts of sample space and event space.

Short Answer

Expert verified
The sample space is all possible outcomes; the event space is a subset of outcomes meeting specific criteria.

Step by step solution

01

Understand Sample Space

The sample space of an experiment is the set of all possible outcomes. For instance, in a dice roll, the sample space is \( \{1, 2, 3, 4, 5, 6\} \) because each side of a fair six-sided die is an equally possible outcome.
02

Define Event Space

An event space, also known simply as an event, is a subset of the sample space. It consists of outcomes that satisfy a certain condition. For example, if we are interested in rolling an even number on a dice, the event space is \( \{2, 4, 6\} \).
03

Distinguish Between Sample and Event Spaces

The key difference between a sample space and an event space is that the sample space contains all possible outcomes, while an event space includes only those outcomes that precisely meet the criteria of the event in question. In any experiment, the sample space remains constant, but different criteria might lead to different event spaces.

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Key Concepts

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

Sample Space
To begin with, let's talk about the concept of the sample space, an essential part of probability theory. The sample space is defined as the set of all possible outcomes in a particular experiment.
Think of it as the complete list of every possible result that can occur. It is denoted typically by the symbol \( S \).
For example, when you roll a six-sided die, you can end up with one of the following results: 1, 2, 3, 4, 5, or 6.
Therefore, the sample space for this experiment is \( \{1, 2, 3, 4, 5, 6\} \).
  • The sample space is fundamental because any probability calculation is based on it.
  • It's the complete set and nothing beyond these outcomes can occur.
In essence, understanding the sample space is like viewing the entire board on which all outcomes can play out in your experiment.
Event Space
Now that we understand the sample space, let's explore the concept of an event space, often simply called an event.
An event space is a subset of the sample space, containing all the outcomes that meet a specific condition. For instance, if we are interested in rolling an even number on a six-sided die, our event space will only include those outcomes that are even numbers.
So, in this example, the event space is \( \{2, 4, 6\} \). This particular event is a part of the larger sample space, \( \{1, 2, 3, 4, 5, 6\} \).
The following points clarify an event space further:
  • It can consist of one, multiple, or none of the outcomes from the sample space.
  • Think of it as filtering the sample space according to the condition set by the event.
The event space allows us to zero in on the specific outcomes we care about for our probability questions.
Outcomes in Probability
Finally, let's bring it all together by discussing outcomes in probability, which form the building blocks of both the sample and event spaces. An outcome is a single result of an experiment.
In our dice example, rolling a "2" is one such outcome. When you collect all possible outcomes, you get the sample space.
  • All outcomes in the sample space are equally possible, especially in fair experiments like a dice roll or a coin toss.
  • When you isolate certain outcomes to form an event space, you are focusing on outcomes of interest.
Understanding outcomes is crucial because probabilities are assigned to these outcomes or combinations thereof. Thus, outcomes are the fundamental components enabling probability calculations for any experiment.

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