The **sample space** (denoted ) is defined as the set of all possible distinct outcomes or “events” of an experiment. Using this perspective, a classic idea about ‘probability’ is

provided all points in are equally likely.

**Definition** Let be a discrete sample space. Then the **probabilities** are numbers attached to the such that two conditions hold:

(1)

(2)

**Definition** An event in a discrete sample space is a subset . If the event contains only one point, e.g. we call it a simple event. An event A made up of two or more simple events, e.g. is a compound event.

**Definition** The probability of an event is the sum of the probabilities for all the simple events that make up .

A random variable is a numerical-valued variable that represents the outcomes in an experiment or random process. Typically, random variables are denoted by an upper-case letter such as . The corresponding lower case letter is often reserved to refer to one of a number of possible values that the random variable can take on. For example, if a coin is tossed 3 times then

**Definition** A **random variable** is a function that assignes a real number to each point in a sample space .

One type of desired description of a random variable is a summary for how probability is distributed amongst the possible values a random variable can take on.

**Definition** The **probability function** of a random variable is the function

The pairs is collectively the **probability distribution**. All probability functions share two properties:

(1)

(2)

An alternative to the probability distribution for describing a probability model is the **cumulative distribution function** or simply the **distribution function**.

**Definition** The **distribution function** of a random variable is

All distribution functions share three properties:

(1)

(2)

(3)