Mathematics Probability Theory is a branch of mathematics that deals with the analysis and interpretation of random phenomena. It provides a framework for quantifying uncertainty, allowing for the calculation of the likelihood of various outcomes based on known information. Probability Theory is foundational for various fields, including statistics, finance, science, and engineering, and encompasses concepts such as random variables, probability distributions, expectation, and statistical inference. Through its principles, it enables the modeling and understanding of complex systems where uncertainty exists.

Fundamental Concepts of Probability Theory Random Experiments Definition and Characteristics Unpredictability Repeatability Outcome Set Examples Tossing a coin Rolling a die Drawing a card Sample Space Definition Set of all possible outcomes Types of Sample Spaces Discrete Sample Space Finite number of outcomes Countable outcomes Continuous Sample Space Uncountable outcomes Example: Range of real numbers between 0 and 1 Representation List notation Set builder notation Venn diagrams Events Definition Subset of the sample space Types of Events Simple Events Events with a single outcome Example: Getting a head when a coin is tossed Compound Events Events formed by combining two or more simple events Use of logical operations (union, intersection) Mutually Exclusive Events Events that cannot occur simultaneously Example: Rolling an odd and an even number in a single die throw Exhaustive Events A set of events covering the entire sample space Every possible outcome is included Operations on Events Union of Events Definition and notation Example using Venn diagrams Intersection of Events Definition and notation Example using Venn diagrams Complement of an Event Definition and notation Example using Venn diagrams Probability Axioms Non-negativity Probability of any event is a non-negative number Additivity For mutually exclusive events, probability of their union is the sum of their individual probabilities Normalization Total probability of the entire sample space equals one Properties Derived from Axioms Monotonicity Probability of any event is less than or equal to one Complement Rule Probability of an event plus the probability of its complement is one Inclusion-Exclusion Principle Relationship to calculate probability of the union of two events Real-world Applications Weather Forecasting Modeling uncertainty in predictions Quality Control Assessing probabilities of defects in manufacturing Epidemiology Estimating the likelihood of disease outbreaks Limitations and Paradigms Classical Interpretation Based on symmetry and equally likely outcomes Frequentist Interpretation Long-run relative frequency of an event Subjective Probability Based on personal belief or opinion