Contemporary Mathematics

# Key Concepts

### 7.1The Multiplication Rule for Counting

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

### 7.2Permutations

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

### 7.3Combinations

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

### 7.4Tree 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.

### 7.5Basic 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.

### 7.6Probability 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.

### 7.7What Are the Odds?

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

### 7.8The 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.

### 7.9Conditional 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.

### 7.10The 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.

### 7.11Expected 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.
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