Terminology
Although there are many different situations involving randomness, it is convenient to use a general terminology that can cover them all.
Definition
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.
Temperature at midday tomorrow
The outcomes here are the different possible temperatures, such as exactly 15ºC.
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 = A ∩ B = {♥K}
A or B = A ∪ B = {♥A, ♥2, ♥3, ♥4, ♥5, ♥6, ♥7, ♥8, ♥9, ♥10, ♥J, ♥Q,♥K, ♣K, ♦K, ♠K}
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}