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Probability

From sample spaces to the Central Limit Theorem.

Prerequisites: Calculus lessons 1–6

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  1. 1Sample Spaces and EventsThe formal setup of probability: sample spaces, events, and the three axioms.
  2. 2Counting PrinciplesPermutations, combinations, and when order matters.
  3. 3Conditional ProbabilityUpdating probabilities on new information, independence, and the two-children puzzle.
  4. 4Bayes' TheoremReversing conditional probabilities: priors, likelihoods, posteriors, and the medical test puzzle.
  5. 5Random Variables and ExpectationPMFs, CDFs, expected value, variance, and linearity of expectation.
  6. 6Common Discrete DistributionsBernoulli, binomial, geometric, and Poisson: when each applies and their main formulas.
  7. 7Continuous DistributionsDensities, uniform and exponential distributions, and the normal curve.
  8. 8Joint DistributionsJoint, marginal, and conditional distributions; covariance and correlation.
  9. 9LLN & Central Limit TheoremWhy averages settle down, and why everything becomes a bell curve.
  10. 10Infinite ExpectationsA coin game with an infinite expected payout, partial sums as a walking dot, why the harmonic series diverges, and what breaks when E[X] does not exist.
  11. 11Inequalities: Markov, Chebyshev and JensenHow far a random variable can stray when you know only its mean or its variance, and why E[f(X)] differs from f(E[X]) in a direction you can predict, which is where volatility drag and option convexity come from.
  12. 12Symmetry and Geometric ProbabilityAnswer probability questions by spotting interchangeable outcomes and measuring areas: dice orderings, the extra-coin game, meeting problems, the broken stick and points on a circle, mostly without integrals.
  13. 13Order Statistics: The Biggest of ManyThe distribution of the max, the min and the k-th smallest of n draws, the n+1 spacings that uniform points cut a stick into, and how slowly the best of many normals grows.
  14. 14Generating FunctionsMoment and probability generating functions: reading off moments, E[exp(X)] for a normal, adding independent variables by multiplying, dice totals as polynomials, and random sums.

Practice

The problem bank has 61 interview problems on this material, with hints and full solutions.

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