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Concept

Law of large numbers

The statement that the average of independent repetitions of a measurement settles on the underlying mean as their number grows. It is what makes an average an estimate at all, and it says nothing about how fast — that is a separate and finer question.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

One set of sums, two scalings, two different limits. The exact distribution of a sum of n independent copies, scaled two ways. Divided by n it collapses onto the mean; divided by the square root of n it holds a fixed width and settles into a shape.

The average settles and the wobble does not

Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.

probability · Central limit How fast each way of averaging closes in, as the dimension grows. Relative error against the number of points, both on logarithmic scales, for a regular grid in 1, 4, 8 dimensions and for random points in 8; the grid's lines steepen or flatten with the dimension and the random one does not move from a slope of a half.

The error that does not care how many dimensions

A grid gets rapidly better in one dimension and hopelessly worse in twenty. Random points get better at the same slow rate whatever the dimension, which is why a method that is bad everywhere ends up being the only one that works.

probability · Monte Carlo A polygon of 4 points averaged down to its curve at t = 0.40. A control polygon of 4 points, the 3 rounds of weighted averaging at t = 0.40 drawn as nested polylines, the single point they end at, and the whole curve those points trace. The weights on the control points are 0.216, 0.432, 0.288, 0.064.

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.

discrete · Pascals triangle A mass split 10 times, 0.3 to the left and 0.7 to the right. A self-similar measure on the unit interval: the mass is split 10 times, 0.3 of each piece's share going left and 0.7 right, and each of the 1024 pieces is drawn as a bar as tall as its mass. The heaviest carries 0.0282 and the lightest 5.9 × 10⁻⁶.

A dimension for every rate of crowding

Spread a unit of mass over an interval by splitting it unevenly, again and again, and the result covers the whole interval while crowding almost all of its weight onto a set of smaller dimension. Every rate of crowding picks out its own set of points with its own dimension, and the whole family is read off one curve.

dynamics · Fractal dimension Two children, and at least one is a boy. Four equally likely families drawn as quarters of a square: the question “is at least one a boy?” rules out only the girl–girl family, and leaves three equal quarters; the chance of two boys is 33.3%.

Two children and the sentence about one of them

A family has two children and at least one is a boy. The chance that both are boys is one in three — or one in two, or anything from one in three to certainty — and every one of those answers is right for some way the sentence could have come to be said. There is no host and no door, and the protocol is still the whole problem.

probability · Bayes One experiment, measured by runs and by awakenings. Two unit squares for the same coin and the same schedule: the same experiment weighed two ways: by runs, heads keeps half the square; by awakenings, heads is one of 3 equal slices.

One coin, counted by runs and by wakings

Beauty is put to sleep and a fair coin is tossed. Heads, she is woken once; tails, twice, with the first waking erased from her memory. Each time she wakes she is asked how likely heads is. One half, say some; one third, say others; and unlike every earlier puzzle of this kind, stating the protocol exactly does not end the argument.

probability · Bayes A random pairing of 24 edges. Chord diagram of one uniformly random pairing of a 24-gon's edges, corners coloured by the vertex they become. a 24-gon with its edges paired at random and glued head to tail: the 24 corners fall into 3 vertices, so the surface has genus 5, against a most possible of 6.

The surface a random gluing makes

Pair the edges of a large polygon at random and glue each pair head to tail. The surface almost always has nearly as many handles as the polygon allows: a thousand edges leave about seven and a half vertices, and the genus is within four of its ceiling of 250. The vertices behave like the cycles of a random permutation, and their average is a harmonic number.

topology · Surface classification A majority of independent voters, more often right than any of them. The probability that a simple majority of n independent voters is right, for n from 1 to 201, when each voter is right with probability 0.45, 0.51, 0.55, 0.6, 0.7.

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.

applied · Voting rules Records in a thousand draws, and the lengthening waits between them. A sequence of 1000 uniform draws on a logarithmic position axis, with records at positions 1, 6, 11, 27, 36, 145, 539, 927.

A wait that ends and has no average

Draw a number, then keep drawing until one beats it. Half the time the very next draw does. The wait is certain to end, it is usually short, and its average is infinite — and the chance of still waiting after n draws is exactly 1/(n + 1), whatever the numbers are drawn from, because symmetry is the only thing the calculation uses.

probability · Expectation Four fractions staked on one run of tosses. p=0.6, 1000 tosses, 604 heads; final log10 wealth: Kelly 9.449, half Kelly 6.881, twice Kelly 0.409, everything ruined at toss 10.

The best average is the worst bet

A coin lands heads six times in ten and pays even money. Stake everything on every toss and the expected wealth grows by a fifth a toss — and the bettor is ruined at the first tail, which is certain to come. Stake a fifth of wealth each time instead and the average grows far more slowly, while the typical bettor grows faster than with any other fraction. The expectation is maximised by the strategy almost nobody should use.

probability · Expectation

Named alongside it

The objects these essays reach for when they reach for this one.

ExpectationBayes' theoremIndependenceAreaBinomial coefficientBinomial distributionCentral limit theoremConditional probabilityConvergence rateCounting argumentHarmonic seriesLikelihood

All concepts