Probability Theory and Statistics

Probability theory provides a mathematical framework for reasoning about uncertainty by assigning numbers between zero and one to events according to how likely they are. Foundational ideas include the sample space of all possible outcomes, the rules for combining probabilities of independent or mutually exclusive events, and conditional probability, which updates a likelihood once additional information is known. These principles underlie everything from simple dice problems to risk assessment and statistical inference.

Counting the number of ways outcomes can occur is central to discrete probability, and this is the province of combinatorics. Permutations count arrangements in which order matters, while combinations count selections in which it does not. The binomial coefficients that arise in combinations also generate the entries of Pascal's triangle and the terms of the binomial theorem, which expands a power of a sum into a series of products. These counting tools make it possible to compute the probabilities of events involving many equally likely outcomes.

Statistics applies probability to real data. Descriptive measures summarise a data set's centre and spread, while inferential techniques use samples to estimate population characteristics and to test claims. Correlation measures the strength of association between two variables, and regression fits a model — often a straight line — that predicts one variable from another. Together these methods support data analysis across the sciences, economics and engineering, turning raw observations into quantified conclusions.

Frequently asked questions

What is the difference between a permutation and a combination?

A permutation counts ordered arrangements, so different orders are counted separately. A combination counts selections where order does not matter.

How does Pascal's triangle relate to the binomial theorem?

The rows of Pascal's triangle are exactly the binomial coefficients that appear when a power of a sum is expanded, so the triangle lists those coefficients directly.

What is the difference between correlation and regression?

Correlation measures how strongly two variables move together, while regression builds a model that predicts the value of one variable from another.