Probability theory provides the foundation for analyzing random phenomena. This article explores essential concepts like sample space, events, and various event types. We will also dive into set operations like union, intersection, and complement that are used to manipulate events.
1. Sample Space
The sample space is the set of all possible outcomes of a random experiment.
It is denoted by S. The sample space serves as the foundation for understanding
probability since all events are subsets of this space.
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Example: When tossing a coin once, the sample space is:
S = {Heads, Tails} -
Example: When rolling a die, the sample space is:
S = {1, 2, 3, 4, 5, 6}
2. Event Types
In probability theory, an event is a subset of the sample space. Events can be classified into different types:
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Simple Event:
An event that consists of a single outcome from the sample space. Example:E = {Heads}when tossing a coin. -
Compound Event:
An event that consists of more than one outcome from the sample space. Example:E = {2, 4, 6}when rolling a die (the event of getting an even number). -
Certain Event:
An event that will definitely happen. Example: The event of getting a result when rolling a die. -
Impossible Event:
An event that cannot happen. Example: The event of getting a 7 when rolling a standard six-sided die.
3. Set Operations on Events
We can perform several set operations on events, which help in combining or modifying events:
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Union of Events (A ∪ B):
The event that at least one of the events occurs. Example:A = {Heads}andB = {Tails}when tossing a coin. The union of events isA ∪ B = {Heads, Tails}. -
Intersection of Events (A ∩ B):
The event that both events occur. For example, ifA = {Even}andB = {> 4}when rolling a die, the intersection is:A ∩ B = {6}. -
Complement of an Event (AC):
The event that A does not occur. Example: IfA = {Heads}when tossing a coin, the complement isAC = {Tails}.