Ch 4 Introduction to Probabilities

Probability

A numerical value ranging from 0 to 1. It indicates the chance, or likelihood, of a specific event occurring. If there is no chance of the event occurring, the probability is 0. If the event is absolutely going to occur, the probability of it occurring is 1.

Experiment

The process of measuring or observing an activity for the purpose of collecting data. An example is rolling a single six-sided die.

Sample space

All the possible outcomes, or results, of an experiment. The sample space for our single-die experiment is the numbers {1, 2, 3, 4, 5, 6}. Statisicians put the numbers in braces to signal that they represent the sample space.

Event

One or more outcomes of an experiment. The outcome, or outcomes, is a sub-set of the sample space. An example of an event is rolling a pair with two dice.

Simple event

An event with a single outcome in its most basic form that cannot be simplified. An example is rolling a five with a single die.

Classical probability

A probability that is determined by dividing the number of possible outcomes of an event by the total number of possible outcomes in the sample space.

Simple probability

The likelihood of a single (simple) event occurring by itself.

Collectively exhaustive

Describes a set of events where at least one of the events must occur during an experiment.

Empirical probability

A probability determined by counting the frequency of an event during an experiment.

Law of large numbers

A law that states that when an experiment is conducted a large number of times, the empirical probabilities of the process will converge to the classical probabilities.

Subjective probability

A probability that is used when classical and empirical probabilities are not available and is calculated based on experience and intuition.

Complement (to Event A)

Denoted as A'; defined as all the outcomes in a sample space that are not part of Event A.

Contingency table

A table that shows the number of occurrences of events of an experiment classified by two categorical variables.

Marginal probabilities

Simple probabilities found in the margins, or row and column totals, of a contingency table.

Intersection (of Events A and B)

The number of instances in which Events A and B occur at the same time.

Joint probability

The probability of the intersection of two events.

Union (of Events A and B)

All of instances where either Event A or B occur or both events occur together.

A mathematical rule used to calculate the probability of the union of events, that is, the probability that Event A or Event B, or both events will occur.

Mutually exclusive events

Events that cannot occur at the same time during the experiment.

Conditional probability (of A given B)

The probability of Event A occurring, given the condition that Event B has already occurred.

The probability of Event A occurring as determined without any additional information that could influence the event.

Prior probability

Posterior probability

A probability that is a revision of a prior probability using additional information.

Independent events

Events that have no impact on the probability of each other occurring.

Dependent events

Events wherein the occurrence of one event affects the occurrence of another event.

Multiplication rule

A rule used to determine the probability of the intersection (joint probability) of two events occurring, or P(A and B).

Decision trees

Display marginal and joint probabilities from a contingency table.

Fundamental counting principle

A principle that states that if there are k1 choices for the first event, k2 choices for the second event, ... and kn choices for the nth event, then the total number of possible outcomes is (k1)(k2)(k3)...(kn).

Combinations

The number of different ways in which objects can be arranged without regard to order.