Discrete Probability Distributions

Introduction to Probability Distributions

Probability distributions are fundamental tools in statistics for describing the likelihood of various outcomes for a random variable. In the context of discrete probability distributions, the random variable takes on countable values, and each value has an associated probability.

• Discrete random variables represent counts or distinct outcomes (e.g., number of classes taken).

• A probability distribution for a discrete variable is typically presented as a table listing all possible values of the variable and their corresponding probabilities.

• For a probability distribution to be valid:

  • All probabilities must be between 0 and 1.

  • The sum of all probabilities must equal 1.

Example Table: Probability Distribution

Justification: All probabilities are between 0 and 1, and their sum is 1.

Expected Value of a Probability Distribution

The expected value (or mean) of a discrete probability distribution represents the long-run average outcome if the experiment is repeated many times. It is a weighted average, where each value is weighted by its probability.

• Formula:

• Values of x are not equally likely; more probable values contribute more to the expected value.

Example Calculation

Given the table above:

Interpretation: The expected number of classes taken is 4.21. While it is not possible to take a fractional number of classes, the expected value represents the average over many students.

Application: Expected Profit Calculation

Probability distributions can be used to determine expected profit in business scenarios, such as choosing price points for event tickets.

• Suppose there are three seat types: Premium ($100), Middle ($50 or $35), Cheap ($20).

• Probabilities are assigned based on survey data for each seat type.

Example Tables: Price Point Scenarios

• Expected profit for first scenario:

• Expected profit for second scenario:

• Conclusion: Offering the middle level seats at a discounted rate increases expected profit.

Binomial Distribution

Definition and Conditions

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success.

• There must be two outcomes per trial: success or failure.

• The probability of success () must remain constant for each trial.

• The number of trials () must be fixed in advance.

• Trials must be independent; the outcome of one does not affect another.

Binomial Probability Formula

The probability of observing exactly successes in trials is given by:

• is the binomial coefficient, representing the number of ways to choose successes from trials.

• is the probability of success; is the probability of failure.

Mean (Expected Value) of Binomial Distribution

The expected number of successes in trials is:

Factorial Review

Factorials are used in the binomial coefficient calculation.

• Definition:

• and

• When dividing factorials, common terms cancel out.

Calculating Binomial Probabilities

To find the probability of a range of successes (e.g., "at least k", "at most k"), use the binomial formula for each value and sum the probabilities, or use the complement rule:

• At least k:

• At most k:

• Complement rule:

Example: Lottery Tickets

• Suppose you buy 5 lottery tickets, each with a 0.10 probability of winning. Each ticket is independent.

• Probability that none win:

• Probability that exactly one wins:

• Probability that at least one wins:

• Probability that at most one wins:

• Expected number of winning tickets:

Summary Table: Binomial Distribution Properties

Additional info: StatCrunch and other statistical software can be used to compute binomial probabilities efficiently.