Probability: Basic to Advanced

The subject where setup is everything

Probability is unusual in that most errors are not computational. A student writes down the wrong sample space, or conditions on the wrong event, then computes flawlessly and arrives at a confident wrong answer.

So this course spends disproportionate time on setting problems up and on stating exactly what is being conditioned on — and comparatively little on arithmetic, which is rarely the difficulty.

Intuition is unreliable, and that is the content

Human intuition about chance is genuinely poor, and a probability course that does not confront that directly leaves students able to pass examinations and still reason badly.

The clearest case is the base-rate error. Given a test that is 99 per cent accurate for a disease affecting one person in ten thousand, most people — including many professionals — read a positive result as near-certainty. It is not, because almost all positives come from the vast healthy population. Bayes' theorem makes this precise in two lines, and the unit exists because the error is so persistent and so consequential outside the classroom.

Linearity of expectation

If a student takes one thing from this course, it should be this.

Expectation is linear whether or not the variables are independent. That is a remarkable statement and students routinely fail to appreciate what it licenses: a problem that appears to demand tracking complicated dependencies between events can collapse into a sum of individually trivial expectations.

Competition problems are built on this repeatedly. Students who have internalised it start problems that others cannot begin.

The three stages

Basic covers the counting probability needs, sample spaces and events as sets, the axioms, inclusion-exclusion, then conditional probability, independence, the law of total probability and Bayes.

Intermediate covers random variables and expectation, the standard discrete distributions — Bernoulli, binomial, geometric, negative binomial, Poisson, hypergeometric — continuous variables with density and distribution functions, the uniform, exponential and normal distributions, and joint distributions with covariance and convolution.

Advanced covers the laws of large numbers and the central limit theorem, Markov chains including absorbing states and the fundamental matrix, and an introduction to the measure-theoretic foundations.

What the limit theorems do not say

The central limit theorem is probably the most misquoted result in mathematics. It does not say that everything is normally distributed, it does not apply regardless of conditions, and it says nothing useful about the tails at the sample sizes people habitually invoke it for.

We teach the conditions alongside the statement, and we spend time on what these theorems do not claim — partly because that is where they are misused in practice, and partly because a student who knows the limits of a theorem understands it better than one who knows only its conclusion.

Markov chains

Transition matrices, classification of states as recurrent, transient or absorbing, stationary distributions, and absorbing chains with the fundamental matrix and expected time to absorption.

This unit pays off well beyond probability itself: it is the standard model for retention and churn problems, it appears directly in the Integral Cup probability track, and it is the foundation for the MCMC methods that modern statistics runs on.

Where elementary probability stops working

The final unit is an honest signpost. Elementary probability assumes you can assign a number to every event, and once the sample space is uncountable that assumption quietly fails.

Sigma-algebras, measurable functions, probability as a measure, expectation as an integral, conditional expectation as a projection — an introduction rather than a full course, placed last, and optional for students who want the competition material only. Students continuing to graduate study or into quantitative finance will meet the rigorous version regardless, and meeting it once here makes that transition far less abrupt.

Who this suits

Students from Class 9 upward through undergraduate study, with school algebra. The counting this course needs is taught within it; the dedicated combinatorics course goes considerably further for students who find counting is their real obstacle.

Classes are live and online in groups of at most eight, with written work corrected individually.