Event (probability theory)

In probability theory, an event is a set of outcomes of a random experiment to which a probability is assigned. Formally, an event is a subset of the sample space.

Definition

Let (Ω, F, P) be a probability space, where:

• Ω is the sample space (the set of all possible outcomes).
• F is a sigma-algebra of subsets of Ω (the set of events; it must contain Ω itself, the empty set, and be closed under complementation and countable unions and intersections).
• P is a probability measure, which assigns a probability to each event in F.

An event is any element of the sigma-algebra F. In simpler terms, an event is a collection of possible outcomes from the sample space that we are interested in. The probability measure P assigns a value between 0 and 1 (inclusive) to each event, representing the likelihood that the event will occur.

Types of Events

• Simple Event: An event consisting of only one outcome.
• Compound Event: An event consisting of two or more outcomes.
• Certain Event: The sample space Ω itself is an event, and it is a certain event because one of the possible outcomes in Ω must occur. The probability of a certain event is always 1.
• Impossible Event: The empty set Ø is an event, representing an event that cannot occur. The probability of an impossible event is always 0.
• Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. Mathematically, P(A ∩ B) = P(A)P(B).
• Mutually Exclusive Events (Disjoint Events): Two events A and B are mutually exclusive if they cannot occur at the same time. That is, their intersection is the empty set: A ∩ B = Ø. For mutually exclusive events, P(A ∪ B) = P(A) + P(B).
• Complementary Event: The complement of an event A, denoted by A^c or A', is the set of all outcomes in the sample space that are not in A. The probability of the complement is P(A^c) = 1 - P(A).

Examples

Consider the experiment of rolling a six-sided die. The sample space is Ω = {1, 2, 3, 4, 5, 6}.

• Example 1: The event "rolling an even number" is the set {2, 4, 6}.
• Example 2: The event "rolling a number greater than 4" is the set {5, 6}.
• Example 3: The event "rolling a 7" is the impossible event Ø.
• Example 4: The event "rolling a number between 1 and 6 (inclusive)" is the certain event Ω.

Importance

The concept of an event is fundamental to probability theory as it allows us to quantify the likelihood of specific occurrences or sets of occurrences within a random experiment. By defining events as subsets of the sample space and assigning probabilities to them, we can make predictions and draw inferences about the underlying random process.