Probability - Axiomatic Approach to Probability

Random Experiment and Sample Space: A random experiment is a process where the outcome cannot be predicted with certainty, but the set of all possible outcomes is known. This set is called the Sample Space, denoted by $S$. Visually, imagine $S$ as a rectangular boundary containing all possible individual outcomes (sample points).

Events: An event is a subset of the sample space $S$. An event $A$ is said to have occurred if the outcome of the experiment is an element of set $A$. In a Venn diagram, an event is depicted as a circle or a closed region inside the sample space rectangle.

Axiomatic Definition of Probability: Let $S$ be the sample space. Probability $P$ is a function that assigns a real number to each event such that: (i) The probability of any event is non-negative ($P(A) \ge 0$); (ii) The probability of the entire sample space is 1 ($P(S) = 1$); (iii) For mutually exclusive events, the probability of their union is the sum of their individual probabilities.

Mutually Exclusive Events: Two events $A$ and $B$ are mutually exclusive if the occurrence of one excludes the occurrence of the other, meaning $A \cap B = \phi$. Visually, these are represented as two separate circles within the sample space that do not overlap or share any area.

Exhaustive Events: A set of events $E_1, E_2, ..., E_n$ are exhaustive if their union covers the entire sample space, i.e., $E_1 \cup E_2 \cup ... \cup E_n = S$. Visually, this means that the combined area of these events fills the entire sample space rectangle.

Complementary Events: For any event $A$, the event 'not $A$', denoted by $A'$ or $A^c$, contains all outcomes in $S$ that are not in $A$. Visually, if $A$ is a circular region, $A'$ is the entire area of the sample space rectangle that lies outside that circle.

Equally Likely Outcomes: If a sample space has $n$ outcomes and each has the same chance of occurring, the probability of each elementary event is $\frac{1}{n}$. If an event $A$ consists of $m$ such outcomes, then $P(A) = \frac{m}{n}$.

Algebra of Events: This involves operations like Union ($A \cup B$, representing 'A or B' or 'at least one'), Intersection ($A \cap B$, representing 'A and B' or 'simultaneous occurrence'), and Difference ($A - B$, representing 'A but not B'). Visually, $A \cup B$ is the total area covered by both circles, while $A \cap B$ is the overlapping shaded region.