Probability: Axiomatic Approach and Events

An experiment whose outcome cannot be predicted with certainty, but all possible outcomes are known in advance. For example, tossing a fair coin or rolling a die.

A possible result of a random experiment. For example, when tossing a coin, 'Head' is an outcome. When rolling a die, '3' is an outcome.

The set of all possible outcomes of a random experiment. It is denoted by 'S'. Each element of the sample space is called a sample point. For example, if a coin is tossed, S = {H, T}. If a die is rolled, S = {1, 2, 3, 4, 5, 6}.

An event is a subset of the sample space. It is a collection of some outcomes of the experiment. For example, in rolling a die, the event 'getting an even number' is E = {2, 4, 6}.

An event which has no outcomes. It is the empty set (∅) and cannot occur. For example, in rolling a die, the event 'getting a number greater than 6' is an impossible event.

An event which contains all outcomes of the sample space. It is the sample space (S) itself and is certain to occur. For example, in rolling a die, the event 'getting a number less than 7' is a sure event.

An event consisting of a single sample point of the sample space. For example, in rolling a die, getting '3' is a simple event.

An event consisting of more than one sample point of the sample space. For example, in rolling a die, getting an 'even number' (E = {2, 4, 6}) is a compound event.

Two events A and B are said to be mutually exclusive if the occurrence of one precludes the occurrence of the other. In set theory terms, their intersection is an empty set (A ∩ B = ∅). They cannot occur simultaneously. For example, in rolling a die, 'getting an even number' (E={2,4,6}) and 'getting an odd number' (O={1,3,5}) are mutually exclusive events.

A set of events E1, E2, ..., En is said to be exhaustive if their union is the entire sample space (E1 ∪ E2 ∪ ... ∪ En = S). This means that at least one of these events must occur. If events are both mutually exclusive and exhaustive, then P(E1) + P(E2) + ... + P(En) = P(S) = 1.

This approach defines probability based on a set of axioms (postulates) or rules. For a sample space S and an event E, the probability P(E) must satisfy the following three axioms:

The complement of an event A, denoted by A' or Aᶜ, consists of all outcomes in the sample space S that are not in A. The sum of the probability of an event and its complement is 1.

For any two events A and B, the probability of their union (A or B occurring) is given by the sum of their individual probabilities minus the probability of their intersection (both A and B occurring).