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Key Concepts

Key Concepts

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.