Probability
Named by 15 essays across 5 fields — each of them below, with the objects they name alongside it.
The constant that counts what does not happen
Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.
The chain that stops
Give a chain a state it cannot leave and there is no long run to find — every walk ends. What is worth computing instead is how long it lasts and where it finishes, and both are exact answers to a linear system rather than limits of anything.
The chain that runs the same backwards
Put weights on the edges of a graph, step to a neighbour in proportion to them, and the long-run share of a state is its own weight over the total — read straight off the picture, with nothing to solve. The condition that makes that work is strictly stronger than being stationary.
The time spent and the share held
Stationary shares are a limit of distributions — where the walk probably is after many steps. Here the question is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.
Where the shares have nowhere to go
On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.
One table, two lotteries
A table of shares says what fraction of each task each person does. It does not say how — the same table is a mixture of whole assignments in many different ways, and the differences are exactly what the people being assigned would care about.
Nearly always, or nearly never
Toss a coin for every pair of points and ask whether the graph that results has some property. For a property a first-order sentence can state, the answer in the limit is never a genuine probability — it is zero or it is one, and the game is what proves it.
Infinitely many guessers, finitely many wrong
An infinite line of people each wears a black or white hat, sees every hat in front and none of their own, and must guess their own colour. With a finite line, each guesser is right half the time whatever they agree in advance. With an infinite line and the axiom of choice, they can agree a strategy under which all but finitely many are right — and nobody can carry it out.
Fourteen people within a day
Twenty-three people probably include two with the same birthday. Fourteen probably include two whose birthdays are at most a day apart, and seven, two within a week. The near miss has an exact formula, found by a trick that takes k days away after every birthday and turns the question back into the plain one on a shorter year, and the pattern behind every threshold is a single square root: a window of k days either side makes each pair 2k + 1 times as likely to collide.
The ground a walk covers
A random walk of a thousand steps visits far fewer than a thousand places: in one dimension about fifty, in the plane about four hundred, in space about six hundred and sixty. The share of steps that land on new ground is exactly the chance of never coming home — so the number that decides whether a walker returns also decides how much of the world it sees.
A dust that almost every line misses
Keep the four corner squares of a square, then the four corners of each of those, and so on. What is left has a length, in the sense that measures length, and yet almost every straight line misses it entirely. Stage by stage the chance that a random line hits the dust falls — to 48% by the eighth stage — and how fast it falls to nothing is one of the few questions about a simple picture that is still open.
Five-eighths of the pairs, and no more
Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.
Two random shuffles reach every shuffle
Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.
Nineteen copies and no more
Two of the icosahedron's rotations, chosen at random, generate all sixty of them nineteen times in thirty. Philip Hall found the number in 1936 by inclusion and exclusion over every subgroup at once, weighted by the Möbius function of the subgroups' order — and the same count says that nineteen copies of the group, side by side, can still be generated by two elements, while twenty cannot.
Every premise makes the verdict vote stricter
A court that must find several things true before it finds liability can vote on each of them or on the verdict. With two elements, judges right seven times in ten are enough for either. With five, the verdict vote needs judges right eighty-seven times in a hundred, or a larger court makes it worse — while the premise vote still needs only better than even. And when any one ground is enough instead of all, the two procedures swap their failures.
Named alongside it
The objects these essays reach for when they reach for this one.
Random walkExpectationMarkov chainInvariantLimitStationary distributionAlternating groupCounting argumentExhaustive searchGenerating setInclusion exclusionParity