7.12.1: Key Concepts

Contemporary Mathematics · LibreTexts

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    7.1 The Multiplication Rule for Counting

    • The Multiplication Rule for Counting is used to count large sets.

    7.2 Permutations

    • Using the Multiplication Rule for Counting to enumerate permutations.
    • Simplifying and computing expressions involving factorials.
    • Using factorials to count permutations.

    7.3 Combinations

    • Permutations are used to count subsets when order matters; combinations work when order doesn't matter.
    • Combinations can also be computed using factorials.

    7.4 Tree Diagrams, Tables, and Outcomes

    • We identify the sample space of an experiment by identifying all of its possible outcomes.
    • Tables can help us find a sample space by keeping the possible outcomes organized.
    • Tree diagrams provide a visualization of the sample space of an experiment that involves multiple stages.

    7.5 Basic Concepts of Probability

    • The theoretical probability of an event is the ratio of the number of equally likely outcomes in the event to the number of equally likely outcomes in the sample space.
    • Empirical probabilities are computed by repeating the experiment many times, and then dividing the number of replications that result in the event of interest by the total number of replications.
    • Subjective probabilities are assigned based on subjective criteria, usually because the experiment can’t be repeated and the outcomes in the sample space are not equally likely.
    • The probability of the complement of an event is found by subtracting the probability of the event from one.

    7.6 Probability with Permutations and Combinations

    • We use permutations and combinations to count the number of equally likely outcomes in an event and in a sample space, which allows us to compute theoretical probabilities.

    7.7 What Are the Odds?

    • Odds are computed as the ratio of the probability of an event to the probability of its compliment.

    7.8 The Addition Rule for Probability

    • The Addition Rule is used to find the probability that one event or another will occur when those events are mutually exclusive.
    • The Inclusion/Exclusion Principle is used to find probabilities when events are not mutually exclusive.

    7.9 Conditional Probability and the Multiplication Rule

    • Conditional probabilities are computed under the assumption that the condition has already occurred.
    • The Multiplication Rule for Probability is used to find the probability that two events occur in sequence.

    7.10 The Binomial Distribution

    • Binomial experiments result when we count the number of successful outcomes in a fixed number of repeated, independent trials with a constant probability of success.
    • The binomial distribution is used to find probabilities associated with binomial experiments.
    • Probability density functions (PDFs) describe the probabilities of individual outcomes in an experiment; cumulative distribution functions (CDFs) give the probabilities of ranges of outcomes.

    7.11 Expected Value

    • The expected value of an experiment is the sum of the products of the numerical outcomes of an experiment with their corresponding probabilities.
    • The expected value of an experiment is the most likely value of the average of a large number of replications of the experiment.

    This page titled 7.12.1: Key Concepts was last modified on Thu, 02 Jan 2025 09:27:32 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.