Chapter VI - Probability

Probability Rule (Total Probability)

The probabilities of all possible outcomes summed together must be equal to one.

Event

Any given outcome or set of outcomes of a given random phenomenon. It is a subset of the sample space.

Probability Rule (Definition of Probability)

Any probability of any given event is a number between zero and one.

Disjoint Events

Events that do not share any outcomes, can never occur simultaneously, and are also known as mutually exclusive. These events are not independent because if one of the disjoint events occurs, the others cannot.

Multiplication Principle

If one can perform an original task in X different ways and a second task in Y different ways, he or she can perform both tasks in (X)(Y) different ways.

Joint Probability

The probability of a joint event.

Random

The descriptor of an order that is unpredicatable in the short term but has a regular pattern in the long run.

Independent

The descriptor that indicates knowing the occurrence of one event does not chance the probability another event will occur. If two events (A and B) have positive probability and the probability of A given B equals the probability of B, they are independent. This concept cannot be displayed on a Venn Diagram.

Tree Diagram

The display of probabilities that obeys the rule stating the probability of reaching the end of any complete branch is the product of the probabilities written on the branch's segments.

Venn Diagram

The display of the sample space that indicates intersection of events.

Intersect

The event that all of a collection of events has occurred, which is best displayed using a Venn Diagram.

Union

The event that at least one of a collection of event occurs.

Empty Event

The event that has no possible outcomes, which can occur in the intersection of two disjoint events.

Complement

The event, denoted X superscript C, that another event (X) does not occur.

Probability Model

The mathematical description of a random phenomenon consisting of the sample space and a way to assign probabilities to events.

Equally Likely Outcome

The probability of any event is equal to the number of outcomes in that event divided by the total number of outcomes in the sample space or the number of outcomes in that event divided by the number of possible outcomes.

Probability in a Finite Sample Space

The probability of any one event is the sum of the probabilities of the outcomes making up the event.

General Addition Rule for Unions of Two Events

The probability of either the first event (A) or the second event (B) occuring is equal to the probability of A plus the probability of B miuns the probability of both A and B occuring. Also, the probability of the union of A and B is equal to the probability of A plus the probability of B minus the probability of the disjointing of A and B.

Bayee's Rule

The probability of one event (A) occuring given that another event (B) has occurred is equal to the quotient of the conditional probability of A given B multiplied by the probability of A and the sum of the conditional probability of B given A multiplied by the probability of A and the conditional probability of B given the complement of A multiplied by the probability of the complement of A.

Conditional Probability

The probability of one event (B) under the condition that another event (A) has occurred. This is calculated as the probability that A and B occur divided by the probability that A occurs.

Probability Rule (Independence)

The probability of two independent events occuring is equal to the product of the probability of one event multiplied by the probability of another event. This is also known as the multiplication rule for independent events.

Probability Rule (Complement)

The probability that any event will not occur is equal to one minus probability that the event will occur.

General Multiplication Rule for Any Two Events

The probability that both of two events (A and B) happen together is equal to the probability of B multiplied by the conditional probability that the A occurs given B has occurred.

Probability Rule (Addition)

The probability that either of two disjoint events will occur is equal to the probability that one event will occur plus the probability that the other will occur.

Sampling Without Replacement

The process through which one changes the independent of outcomes by not maintaining the sample size.

Sampling With Replacement

The process through which one maintains the independence of outcomes by making sure the sample size remains the same.

Probability

The proportion of times the outcome of a random phenomenon will occur in a very long series of repetitions.

Sample Space

The set of all possible outcomes of a given random phenomenon and is denoted S.

Joint Event

The simultaneous occurrence of two events.