Theoretical Probability: Key Concepts and Sample Spaces Explained

Theoretical Probability Vocabulary event: a set of possible outcomes resulting from an experiment; in general, an event is any subset of a sample space experiment: an activity involving chance outcome: result of a trial or experiment probability: the chance or likelihood of an event occurring repeated experiment: a random experiment done with the same conditions and parameters as a previous one sample space: in a probability model for a random process, a list of the individual outcomes that are to be considered of an event occurring single experiment: a random experiment performed once theoretical probability: a number between 0 and 1 representing the likelihood of an event in a theoretical model based on a sample space; if all outcomes in the sample space are equally likely, then theoretical probability of an event is the ratio of the number of outcomes in the event to the number of outcomes in the sample space Sample Spaces page 2 An activity involving chance or probability is called an experiment . . The result of an experiment is an outcome One or more outcomes are called an _ event _. Give an example of an outcome and an event. For example, rolling the number 3 on a number cube, which is one outcome, can be an event. Rolling the number 3 can also be a part of an event like rolling an odd number, where the outcomes are 1,3, and 5. A list of all possible outcomes in an experiment is called the sample space . Single Experime nts Single experiments are experiments that are performed once. Give an example of a single experiment. flipping a coin one time. Think about what could happen when you flip a coin. What are the possible outcomes? There are 2 possible outcomes. The coin could land on heads, or the coin could land on tails. These outcomes make up the sample space, which can be written in set notation or in a list. Set notation may include the letter S for sample space and brackets { }. Outcomes make up the sample space, which can be written in set notation or in a list. Set notation may include the letter S for sample space and brackets { }. Write the possible outcomes of flipping a coin: Set Notation S = {heads, tails} List heads, tails Specifics about Sample Space Fill in the blanks and give an example of each statement.  Outcomes can be written in any order. Example: For example, you may write the sample space for a spinner with the colors red, white, and blue as S = {red, white, blue} or S = {blue, red, white}.  Some sample spaces have outcomes that are the same. If an outcome happens more than once in an experiment, you list each occurrence separately. .

Example: For example, Ravi has non-fiction books, mystery books, fiction books, and biography book on a shelf. The sample space for randomly selecting a book from the shelf can be shown in set notation as S = {non-fiction, non-fiction, mystery, mystery, mystery, fiction, fiction, fiction, fiction, fiction, biography}.  Outcomes can be written using abbreviations. . Example: For example, Ravi's sample space of his books could also be represented as S = {f, n, n, f, m, m, f, f, m, b, f}. Repeated Experime nts Repeated experiments are experiments that are performed two or more times. Give examples of repeated experiments. Some examples include flipping a coin four times, tossing a number cube twice, or spinning a spinner two times. List all of the possible outcomes of playing a video game two times. There are 4 possible outcomes in the sample space: WW, WL, LW, LL You could win both games, win neither game, win the first game and lose the second game, or lose the first game and win the second game. With Replace ment You are using an 8-count box of colored pencils to color a map in history class. Every pencil you pulled out of the box to use, you placed back in the box when finished. Why is this important? Think about when the next class comes in and a student grabs that same box of colored pencils. If you returned the colored pencil to the container, they will have the same options that you did for coloring their map. Explain how replacement affects the sample space. In mathematical terms, they will have the same sample space. When you put the colored pencil back, you returned the sample space to its original state with exactly the same options. In probability, we refer to the action of returning the colored pencil to the box as "with replacement." This idea can be used in repeated experiments. Sample Space of Repeated Experime nts page ___ List ways that the sample space for a repeated experiment can be represented. You can use a tree diagram, a table, a list, or a written description. Learn to Determin e Sample Space A bag contains 3 marbles, 1 red, 1 blue, and 1 green. Determine the sample space of drawing two marbles with replacement.

Learn More About Determini ng Sample Space Review the table in the course to determine the sample space for spinning the spinner twice. There are 36 possible outcomes in the sample space. Important A written description may be a better way to describe a very large sample space. The sample space for spinning the spinner twice can be described as: A written description may be a better way to describe a very large sample space. Because the previous example was performed twice, each outcome has two elements like an ordered pair. It can be described as "The collection of ordered pairs in which each element is either B 1 , B 2 , G, Y, R 1 , or R 2 ."

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