Martingale
Named by 6 essays across one field — each of them below, with the objects they name alongside it.
Two barriers and a fair game
A fair walk between two absorbing barriers is ruined with a probability that is a straight line in the starting stake, and lasts for a number of steps that is the product of what each side can lose. Both facts come from the same two-line recurrence, and both are bad news for the smaller player.
A coin that lets the first player win
On a fair coin the second player in Penney's game always has a better pattern than the first, and the first can hold them to no worse than two to one. Bend the coin and every overlap is paid for in the letters it uses: the replies change, the first player's share swings between a third and a half, and past a heads chance of 1/∛2 the first player simply names HHH and wins.
Three patterns in a circle
Race three coin patterns at once and the gamblers' accounting still gives each one's chance of arriving first — one fairness equation per pattern. What it does not give is any way to read the three-way result off the two-way ones. HHHT, TTHH and HTTH beat one another in a circle, and HHH loses both its head-to-head races and still finishes ahead of one of the patterns that beat it.
Evidence measured in decibans
Write a probability as odds and take the logarithm, and every piece of evidence becomes a length. A positive result on a good test is thirteen decibans; a negative one is minus twenty. Lay the lengths end to end from the prior and the posterior is where they stop, in any order. The rule fails in exactly one way — when two pieces of evidence share a cause — and Turing built a code-breaking method on the arithmetic.
A walk that follows its own footsteps
Let a walk step up with probability equal to the share of up-steps it has taken so far, and every count of up-steps after twenty steps is exactly equally likely. The walk leaves along a straight line at a speed chosen by its first few steps, and although it is expected to come back to its start infinitely often, almost every run comes back only a few times.
Two-thirds of a step past the line
Add random numbers between nought and one until the total passes one, and it takes e of them on average. Ask the same about passing any t and the answer is an exact finite sum that settles, within a few units, onto the line 2t + 2/3. The slope is one over the average step. The two-thirds is the interesting part: it is what the one step that crosses the line charges for being chosen by its length, and it is set entirely by how much the steps vary.
Named alongside it
The objects these essays reach for when they reach for this one.
ExpectationMarkov chainRandom walkConditional probabilityFinite automatonRecurrenceAbsorbing stateBayes' theoremBeta distributionConvergence ratee, the numberExchangeability