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Sample Space in Statistics: Definition & Examples

Sample Space in Statistics: Definition & Examples
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  • 0:02 Sample Space
  • 1:22 Counting Principle &…
  • 2:58 Examples
  • 5:31 Lesson Summary
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Lesson Transcript
Instructor: Ria Yambao

Ria has taught College Algebra and Biostatistics. She has a master's degree in Applied Mathematics.

In this lesson, you will learn the definition of sample space - an important concept in the study of probability. Examples and quiz questions will illustrate how this concept exists in the real world.

Sample Space

You decided to perform a random experiment of rolling a single fair die with six sides. A random experiment is one where you cannot be absolutely sure what the outcome would be prior to performing the experiment. An outcome is a possible result of the experiment. Each side of the die has 1, 2, 3, 4, 5 or 6 dots. What are all the possible outcomes for this experiment? In a single roll of the die, it can show 1, 2, 3, 4, 5 or 6 dots. The exhaustive list of possible outcomes for this experiment is: 1, 2, 3, 4, 5, 6. The set whose members are all possible outcome of a random experiment is called the sample space for that experiment. For our die-rolling experiment, we can write the sample space as:

S = {1, 2, 3, 4, 5, 6}.

An event is a particular set whose members are from a sample space. For example, the event that the die shows an odd number of dots has as its members 1, 3 and 5. We can write

E = {1, 3, 5}.

In this lesson, we will assume that every experiment is a random experiment.

Counting Principle & Tree Diagram

The fundamental counting principle is an important concept that can be used to determine the number of members in a sample space. The fundamental counting principle states that if the first activity can be done in m ways and a second activity can be done in n ways then the first activity followed by the second activity can be done in m times n ways.

Let's discuss the principle using a simple example. Suppose you perform a simple random experiment that involves two steps. First, you toss a coin. Then, you roll a die. How many different outcomes are there for this two-step experiment? In tossing a coin, there are two possible outcomes - Head (H) and Tail (T). In rolling a die, there are six possible outcomes based on the number of dots on each face - 1, 2, 3, 4, 5, 6. The possible outcomes for the two-step experiment are shown in the tree diagram on screen:

Tree Diagram Showing Outcomes of Two-Step Experiment
Diagram Showing Outcomes of Two-Step Expt

A tree diagram is an organized way of writing out all the possible outcomes of an experiment. It is a useful tool in determining the members of a sample space. It can also be used in determining outcomes for an event. Note that since there are two possible outcomes for the coin-tossing activity (step 1) and six possible outcomes for the die-rolling activity (step 2), then there are twelve possible outcomes (2 x 6 = 12) for the two-step experiment.

The fundamental counting principle can be extended to any finite number of activities. When there are three activities:

First Activity Second Activity Third Activity First, Second and Third Activities, in that order
m n k m times n times k

Examples

Let's now go over some examples to put what we've learned into practice.

Example 1. A teacher gave a three-question true or false quiz to her class. If students are not allowed to leave a question unanswered and the only allowed answers for each question are T (for true) or F (for false), how many different possible outcomes are there? What are all the different possible ways of answering the quiz?

Answer:

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