Probability

Tree Diagrams

A tree diagram is a way of describing all the possible outcomes from a series of events A tree diagram is a way of calculating the probability of all the possible outcomes from a series of events.

SIMPLE EVENT

Any event which consists of a single outcome in the sample space is called an elementary or simple event.

The Law of Large Numbers

As an experiment is repeated more and more times, the proportion of outcomes favorable to any particular event will tend to come closer and closer to the theoretical probability of that event

Expected Value

Compute a "weighted average" by multiplying each possible time value by its probability and then adding the products.

COMPOUND EVENTS

Events which consist of more than one outcome are called compound events.

Events Involving "And"

If we multiply both sides of the conditional probability formula by P(A), we obtain an expression for P(A and B). The calculation of P(A and B) is simpler when A and B are independent.

EXPERIMENT

In the study of probability, any observation, or measurement, of a random phenomenon is an experiment.

OUTCOMES

The possible results of the experiment are called outcomes, and the set of all possible outcomes is called the sample space.

Probability Distributions

The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It is also sometimes called the probability function or the probability mass function.

Properties of Probability

The probability of any event lies between 0 and 1. The probability of an impossible event is 0. The probability of an event certain to occur is 1.

Conditional Probability

The probability of event B, computed on the assumption that event A has happened, is called the conditional probability of B, given A, and is denoted P(B | A).

PROBABILITY

The study of probability is concerned with random phenomena. Even though we cannot be certain whether a given result will occur, we often can obtain a good measure of its likelihood, or probability. P denotes a probability. A, B and C denote specific events. P(A) denotes the probability of event A occurring.

Independent Events

Two events A and B are called independent events if knowledge about the occurrence of one of them has no effect on the probability of the other one, that is, if P(B | A) = P(B), or equivalently P(A | B) = P(A).

mutually exclusive events

Two events A and B are mutually exclusive events if they have no outcomes in common. Mutually exclusive events cannot occur simultaneously.

Venn Diagrams

Venn diagrams are graphical representations of events. A rectangle is used to represent the sample space and closed curves are used to represent events (sets) inside the sample space:

subjective probability

can be thought of as a person's degree of confidence that the event will occur.

The classical interpretation of probability

is a theoretical probability based on the physics of the experiment, but does require the experiment to be performed.

empirical probability

is based on long run relative frequencies and is defined as the ratio of the number of observed outcomes favourable to the event divided by the total number of observed outcomes.

SAMPLE SPACE

the set of all possible outcomes is called the sample space.