Binomial distribution
Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.
A bell curve assembled out of coin flips
Drop six hundred balls through a board of pegs, each bouncing left or right at random, and they pile up in a shape that can be predicted precisely. Nothing coordinated them.
A walk that always comes home, until it does not
Step left or right at random, forever, and the walk returns to where it started with certainty. On a grid it also returns. In space it does not, and about a third of walks leave and never come back.
Averaging down the triangle
Change one word in the rule that builds Pascal's triangle — take a share of each entry above instead of adding them — and the triangle stops counting and starts averaging. The same rule then draws smooth curves from polygons and approximates every continuous function by polynomials, at a rate that no amount of smoothness can improve.
The rate a noisy channel allows
A channel that flips one bit in ten can still carry messages with as few errors as anyone likes — at up to 0.531 message bits per transmitted bit, and at no rate above that. The number is Shannon's capacity, 1 − H(p). Repetition reaches reliability only by sending nothing; a code chosen at random gets there at any rate below the limit; and the reason there is a limit at all is a count of how many flip patterns a block of noise can hold.
A majority wiser than its members
Condorcet's other theorem turns voting round: the voters no longer have preferences but judgements about a single fact, each a little more likely right than wrong. Then a simple majority of many of them is almost certainly right — 6,763 voters who are each right 51% of the time make a majority right 95% of the time. The theorem survives voters worse than a coin, if the average is better. It does not survive voters who share their mistakes, and when their skills differ the right rule weighs votes rather than counting them.
Drawn without putting back
Every concentration bound on this shelf assumes the draws are independent. A real sample is not: a pollster does not ring the same person twice, and every ball taken from an urn changes what is left in it. The dependence runs the helpful way. Wassily Hoeffding proved in 1963 that a sample drawn without replacement is at least as concentrated as one drawn with it, for every convex measure of spread at once — and the variance falls by an exact factor that reaches zero when the whole urn is taken.
Which vote finds the truth more often
A court that must decide two premises and their conjunction can vote on the premises or on the conclusion, and the two votes sometimes disagree. Suppose there is a right answer and each judge is more often right than wrong. Then the question has an exact answer: for three judges the conclusion vote is more often right when competence is below √½ and the premise vote above it, and in a large court the premise vote wins at any competence — while the conclusion vote turns out to be a built-in standard of proof.
The element the judges are worst at
A court can vote element by element or on the verdict, and the comparison so far assumed judges equally good at every element. Real judges are surer about the law than about a contested fact. Make one element hard and both procedures sink towards the majority on that element, but they sink differently: the verdict vote sees only the product of the judges' competences and cannot tell an even court from a lopsided one, while the premise vote is ruled by its weakest majority. And when a judge's errors on two elements come from the same misreading of the facts, the gap between the procedures closes.
The middle value has a bell of its own
The central limit theorem is a theorem about sums, and the median of a sample is not a sum of anything. It has a bell all the same: the middle of n draws spreads like 1/(2f√n), where f is the height of the density at the middle — not the variance, which the median never consults. That one number decides when the median beats the average, by a factor of two for a two-sided exponential and outright for the Cauchy law; and where the density is nought at the middle, the bell fails, slowing to a fourth root or splitting in two.
Named alongside it
The objects these essays reach for when they reach for this one.
IndependenceCondorcet jury theoremCorrelationConvergenceHeavy tailsJudgement aggregationLaw of large numbersLimitMajority ruleNormal distributionVarianceApproximation