Experiments

Definition: Experiment

An experiment in probability theory is a process with the following properties:

• The process can occur multiple times;
• All possible outcomes are known and unambiguously defined;
• It is always one and only one of the outcomes which happens.

Definition: Sample Space

The sample space of an experiment is the set of all possible outcomes of said experiment.

NOTATION

Sample spaces are often denoted by $\Omega$.

Example: Flipping a Coin

Flipping a coin is a very simple experiment whose sample space $S$ contains only two possible outcomes - the coin falls heads-up or the coin falls tails-up. Hence, $S$ is just

$$S=\{\mathrm{heads},\mathrm{tails}\}$$

Example: Rolling a Die

Another common experiment is the roll of a single six-sided die. There are six possible outcomes - the number on the die is 1, 2, 3, 4, 5 or 6. Hence, the sample space is

$$S=\{1,2,3,4,5,6\}$$

Example: Flipping Two Coins

Events

Definition: Event

An event is any subset of the sample space of an experiment.

Defined in this way, mathematical events allow us to closely model real-world conditions. However, we need a way to translate between the mathematical formalism of events and their physical reality. Hence, we say that an event has occurred if the outcome of the experiment is an element of the event.

Tip: Union of Events

The union of a collection of events is the event which occurs if and only if at least one of the events in the collection occurs.

Tip: Intersection of Events

The intersection of a collection of events is the event which occurs if and only if all of the events in the collection occur.

Tip: Complement of an Event

The Complement of an event $E$ is the event which occurs if and only if $E$ does not occur.

Definition: Mutual Exclusiveness

A collection of events are mutually exclusive if and only if every intersection between two of the events is empty.