Terminology

Although there are many different situations involving randomness, it is convenient to use a general terminology that can cover them all.

Definition

Experiment
A phenomenon whose result is uncertain will be called an experiment.
Outcomes
An outcome is the result of an experiment that cannot be reduced to simpler results.
Events
An event is a collection of outcomes.
Sample space
The sample space is the collection of all possible outcomes.

Links to set theory

These definitions are closely associated with set theory. Although the ideas behind probability can be understood without it, a formal definition of probability and its rules uses set theory. The links between the definitions above and set theory are shown below:

Probability notation Set theory notation
Sample space, S Universal set
Outcome Element
Event, E Subset

Note carefully the distinction between and event and an outcome. An outcome is indivisible whereas an event might include several possible outcomes.

A card randomly picked from a shuffled deck

In this 'experiment', there are 52 outcomes — the 52 different cards.

Sample space
The sample space is the set of all possible cards, {♥A, ♥2, ♥3, ..., ♥K, ♣A, ..., ♣K, ♦A, ..., ♦K, ♠A, ..., ♠K}
Event
An example of an event is getting a king, {♥K, ♣K, ♦K, ♠K}. It is a set of four outcomes and is a subset of the sample space.

Temperature at midday tomorrow

The outcomes here are the different possible temperatures, such as exactly 15ºC.

Sample space
The sample space is the set of all possible temperatures. For this example, the sample space is infinite since it includes all numbers between say -20ºC and 40ºC.
Event
An example of an event is a temperature between 10ºC and 20ºC. In set theory notation, this could be expressed as {x | 10 < x < 20}.

Set operations

In set theory, unions and intersections of sets are basic operations. These operations also correspond to meaningful ways to define events from others. Consider two events A and B.

Probability notation Set theory notation Interpretation
A or B A ∪ B Either A or B (or both) occurs
A and B A ∩ B Both A and B occur

The complement of a set is also meaningful in terms of events.

Probability notation Set theory notation Interpretation
not A Ac Event A does not occur

A card randomly picked from a shuffled deck

If event A is getting a king, and B is getting a heart,

A = {♥K, ♣K, ♦K, ♠K}

B = {♥A, ♥2, ♥3, ♥4, ♥5, ♥6, ♥7, ♥8, ♥9, ♥10, ♥J, ♥Q, ♥K}

A and B
The event (A and B) corresponds to getting a card that is both a king and a heart, the intersection of the two events,

A and B   =   A ∩ B   =   {♥K}

A or B
The event (A or B) corresponds to getting a card that is either a king, a heart or both, and contains all events in one or other of the sets,

A or B   =   A ∪ B   =   {♥A, ♥2, ♥3, ♥4, ♥5, ♥6, ♥7, ♥8, ♥9, ♥10, ♥J, ♥Q,♥K, ♣K, ♦K, ♠K}

not B
The event (not B) is the set containing all cards that are clubs, diamonds and spades.

Temperature at midday tomorrow

If A is a temperature between 5 and 15ºC, and B is a temperature between 10 and 20ºC,

A = {x | 5 < x < 15}

B = {x | 10 < x < 20}

Then

A and B   =   A ∩ B   =   {x | 10 < x < 15}

A or B   =   A ∪ B   =   {x | 5 < x < 20}

not B   =  Bc  =   {x | x ≤ 10 or x ≥ 20}