Events in Probability Theory
In probability theory, an event E is a subset of the sample space. The event includes only the outcomes that satisfy its conditions.
Example of an Event
When rolling a die, the sample space is a finite set containing 6 elements (possible outcomes):
$$ S = \{ 1, 2, 3, 4, 5, 6 \} $$
An example of an event E is rolling an even number.
The event E is a set containing three outcomes:
$$ E = \{ 2, 4, 6 \} $$
Another event is rolling an odd number.
Event D is also a set with three outcomes:
$$ D = \{ 1, 3, 5 \} $$
Yet another event is rolling a number less than 3.
Event F is a set with two outcomes:
$$ F = \{ 1, 2 \} $$
These three events, D, E, and F, can be represented using a Venn diagram.
Here, S represents the sample space, and the numbered circles represent the outcomes.
The Empty Event
If an event contains no outcomes, it is called an empty event and is denoted by the symbol for the empty set.
For instance, the event F, representing negative numbers in a dice roll, is an empty event because it contains no possible outcomes.
$$ F = \{ \} = Ø $$
Subset and Equality Relations
If all outcomes of event A are also outcomes of event B, we say that A is a subset of B: $$ A ⊂ B $$
The two events are in a subset relationship.
This means that if A occurs, then B will also occur.
Example: Event G consists of all outcomes where the dice roll results in a 2: $$ G = \{ 2 \} $$ Event G is a subset of event E (the even numbers): $$ G ⊂ E $$ that is, $$ \{ 2 \} ⊂ \{ 2, 4, 6 \} $$ As shown in this Venn diagram:
If both events are subsets of each other, this means the two sets have the same outcomes:
$$ A ⊂ B, A ⊃ B \Leftrightarrow A = B $$
In this case, the two events are in an equality relationship.
Union and Intersection of Events
The union of events E and D includes all the outcomes from both E and D. This occurs when at least one of the events E or D happens:
$$ E \cup D = \{ 2, 4, 6 \} \cup \{ 1, 3, 5 \} = \{ 1, 2, 3, 4, 5, 6 \} $$
In this case, the union is equivalent to the entire sample space.
The intersection of events E and F consists of the outcomes that are common to both E and F:
$$ E \cap F = \{ 2, 4, 6 \} \cap \{ 1, 2 \} = \{ 2 \} $$
This can be visualized as follows:
If the intersection of two events is empty, it means the events cannot occur simultaneously. Such events are called disjoint events or mutually exclusive events:
$$ E \cap D = \{ 2, 4, 6 \} \cap \{ 1, 3, 5 \} = \{ \} $$
The Complement of an Event
The complement Fc of an event F is the set of outcomes that do not satisfy F.
Example:
If event F is the set of numbers less than three when rolling a die:
$$ F = \{ 1, 2 \} $$
its complement is the set of numbers greater than two:
$$ F^c = \{ 3, 4, 5, 6 \} $$
This can be shown in a Venn diagram:
And so on.
