Probability

Definition

The probability of an event E is given by:

the number of occurrences of event E divided by the total number of occurrences of all event(s)

for a large number of occurrences.

Probability of an Event not Occurring

The probability that an event A does not occur is given by:

one minus the probability that it happens

Mutually Exclusive Events

Two events are mutually exclusive if they cannot occur simultaneously. The following Venn diagram represents two such events: A and B.

Venn diagram of two mutually exclusive events

In this case the probability of event A or B occurring is given by:

the probability of A plus the probability of B

The OR Rule

The above rule in works for events that are mutually exclusive. Events that are not mutually exclusive can occur simultaneously. In the following Venn diagram, the overlapping region, AB, represents two events occurring at the same time.

Venn diagram of two events that are not mutually exclusive

In this case, the probability of A or B (or both) occurring is given by:

the probability of A plus the probability of B minus the probability of A intersection B

Independent Events

If events A and B are independent than the probability of A occurring does not depend on B having occurred, and vice versa. For two independent events, the probability of both occurring is the product of the individual probabilities:

the probability of A intersection B is the probability of A times the probability of B

Conditional Probability

If two events are not independent, we can define the conditional probability, P(B|A), as the probability of B occurring, given that A has already occurred. The AND rule then becomes:

the probability of A intersection B is the probability of A times the probability of B given  A

This rearranges to:

the probability of B given A is the probability of A intersection B divided by the probability of A

as an equation for conditional probability.

Tree Diagrams

These show the outcomes of a set of events and are useful for probability calculations. The probabilities are written on the branches of the tree. The diagram below is for two events, A and B.

Tree diagram of two two-outcome events

Finally

The above notes are a revision summary rather than a tutorial. In order to understand probability fully, you should now look at some questions.

Valid XHTML 1.0!