Simulation
Named by 3 essays across one field — each of them below, with the objects they name alongside it.
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.
A room where nobody is alone
Twenty-three people probably include two who share a birthday. How many are needed before every single person shares a birthday with somebody else in the room? The answer is 3,064 — more than it takes for every day of the year to be somebody's birthday — and the reason is a count of loners, which rises as the room fills, peaks at 134 when the room is the size of the year, and then falls so slowly that the last loner lingers for thousands of arrivals.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Birthday problemPoisson approximationProbabilityCounting argumentCoupon collectorDimensionExpectationExpected valueGrowth rateMarkov chainRandom walkRecurrence