Reference
Randomness & Fairness Guides
28 reference pages on how random draws actually work — and on the much messier question of getting a group to accept one.
Every tool on this site rests on a small number of mechanisms: an unbiased integer draw, a Fisher-Yates shuffle, a cumulative-sum walk for weighted odds, a derangement for gift exchanges, and a seeded generator for the draws somebody has to be able to check afterwards. The tool pages explain their own mechanic in the context of using them. These guides explain the mechanisms themselves, in general, and are the right place to start if you want to understand rather than to decide something.
The last group is the one people underestimate. A draw can be provably unbiased and still be rejected by the room, and most of the disputes worth writing about are not about the mathematics at all.
How the draws themselves work
The mechanisms every tool on this site is built from. Read these if you want to know what is happening between clicking a button and seeing a result, or if you are implementing something similar yourself.
How Shuffling Actually Works
Fisher-Yates, one swap at a time, and why it is the only shuffle worth using.
Why sort(() => Math.random() - 0.5) Is Biased
The one-line shuffle everyone copies, measured — and the distribution it actually produces.
Modulo Bias and Rejection Sampling
Why % 6 on a random byte quietly favours low numbers, and the two-line fix.
How Weighted Random Selection Works
Cumulative sums, one float, and why duplicating entries is the wrong way to do it.
Constraint Satisfaction in Fair Draws
What happens when the rules and the randomness disagree — and when there is no answer.
Derangements and the Secret Santa Problem
Why a plain shuffle fails a gift exchange 63% of the time, and what a derangement is.
Bracket and Fixture Scheduling Maths
Byes, seeding, and the circle method that fits a full round-robin into n-1 rounds.
How Playlist Shuffle Actually Works
Nobody ships a true shuffle any more. What replaced it, and why it feels more random.
Where the randomness comes from
The generator underneath, and the difference between a number nobody can predict and a number anybody can reproduce. Both are useful; they are useful for opposite reasons, which is the thing most often got wrong.
Pseudo-Random vs Cryptographic Randomness
Two generators, two guarantees, and the surprisingly small list of cases where the difference bites.
Web Crypto getRandomValues: A Reference
What the browser API guarantees, what it does not, and the limits worth knowing.
Seeded and Reproducible Draws
A draw anyone can re-run is a draw anyone can check — and a draw anyone can predict.
Which Randomness Each Tool Here Uses
A tool-by-tool table of the exact draw behind every generator on this site.
Entropy and Password Strength
Bits of entropy, in plain terms, and why a longer random password beats a cleverer one.
Probability you will actually use
The results that come up at a games table, in a spreadsheet, or in an argument about whether something is rigged. No proofs, and no notation beyond what a number needs.
Dice Probability Reference: d4 to d20
Single dice, sums, advantage, and dice pools — the numbers you actually look up.
Expected Value, Explained With Dice
The average of a thing that never happens, and why it is still the right number.
The Law of Large Numbers, Practically
What 'it evens out' actually promises, on what timescale, and what it never promises.
Coin Flip Streaks and What They Mean
How long a run of heads you should expect, as a function of how many times you flip.
The Gambler's Fallacy
The coin does not owe you a tails. Why the intuition is so strong anyway.
The Birthday Problem and Collisions
Why 23 people is enough for a shared birthday, and where else the same maths ambushes you.
Sampling With and Without Replacement
The single decision that changes every number downstream of your draw.
Proving it, and getting it accepted
A draw being fair and a draw being believed are separate problems, and the second one is harder. These cover the evidence, the expectations, and the human failure modes.
How to Prove a Draw Was Fair Afterwards
What evidence actually convinces someone who was not in the room when you drew.
Provably Fair Draws: Commit and Reveal
Publish a hash before the draw, publish the seed after. A worked example, start to finish.
Giveaway Fairness Expectations
What entrants, platforms, and regulators generally expect from a prize draw. General information.
When a Raffle Becomes a Lottery
Consideration, chance, and prize — the three-part test, and why it varies by country. General information.
Why Humans Are Bad at Being Random
Asked to be random, people avoid repeats, alternate too often, and dodge the edges.
How to Audit a Random Tool Yourself
Four checks you can run on any picker in a browser tab, without trusting a word of its copy.
Random Sampling for Audits and Surveys
Sampling frames, sample size, and the mistake that invalidates the whole exercise.
Randomness in the Classroom: A Primer
Where random selection helps a lesson, where it hurts one, and what teachers report.
Looking for a specific situation instead?
These guides explain mechanisms. If you have a concrete job in front of you — a class of thirty to cold-call, five thousand giveaway entries, a wedding table plan with three people who must not sit together — the use-case pages work the other way round, starting from the situation.