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Probability Cards
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Lessons in probability and combinatorics

Short, self-contained lessons with worked examples and exercises. They build on each other, so if you are new to the subject start at the top.

  1. Counting arrangements: permutations

    Why 52 cards can be ordered in more ways than there are atoms on Earth, and how to count ordered selections.

  2. Choosing without order: combinations

    How to count five-card selections, why there are 2,598,960 of them, and how patterns are counted exactly.

  3. Probability from first principles

    Sample spaces, events, equally likely outcomes, complements and the 'at least one' trick.

  4. Drawing without replacement: the hypergeometric distribution

    How many cards of a chosen kind will you see in a draw? A first look at a probability distribution.

  5. Expected value: averages over many repetitions

    Define expected value and use it on neutral, non-wagering examples such as scores, waiting times and survey sampling.

  6. Randomness, shuffles and seeds

    How computers produce random-looking numbers, why seeds make experiments reproducible, and how to shuffle fairly.

  7. Sampling error and the law of large numbers

    Why experiments disagree, how the error shrinks with the square root of the sample size, and what a confidence interval says.

A note on examples

Every example uses cards purely as a clean mathematical model: 52 distinguishable objects in 4 groups of 13. Where a lesson needs a notion of “points” or “scores” to illustrate expected value, they are abstract numbers with nothing at stake. These lessons do not teach or support wagering of any kind.