Probability Theory

Probability theory is the branch of mathematics which studies randomness.

Definition: Randomness

Randomness is an expression of the limits of our knowledge about future events.

The mathematical framework for studying randomness and its properties is built upon the foundations of set theory. We usually frame this study in the terminology of [experiments](Exexperimentsm an experiment, it has a particular outcome. We group all possible outcomes in the so-called [sample space](Esample spacee how often each outcome or a combination of outcomes (also known as an [event](Experimeevent sample space to determine probabilities.

Experiments

Definition: Experiment

An experiment in probability theory is a process whose outcome is always clearly distinguishable from all other outcomes.

Definition: Sample Space

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

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

Definition: Event

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

INTUITION

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

Two events are mutually exclusive iff their intersection is the empty set.