Martingale
Named by 10 essays across 2 fields — 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.
Peeking until the answer is yes
Toss a fair coin, test for bias after every toss, and stop the moment the result is significant at 5%. Within a thousand tosses nearly half of fair coins have been declared biased, and with no limit on the tosses every one of them is. Watch a Bayes factor the same way, stopping at twenty to one, and no amount of watching gets more than one fair coin in twenty past the line.
Evidence that can be multiplied
A p-value from one study and a p-value from the next cannot simply be multiplied, and a researcher who decides whether to run the next study by looking at the last one quietly breaks the rule that combines them. An e-value — a number whose average under the null is at most one — can be multiplied, study after study, under any rule for continuing, and never passes twenty to one more than one time in twenty. What it costs is tosses, and what it demands is a decision about which alternatives to be good at finding.
When to stop tossing a coin
Toss a fair coin as many times as you like and stop whenever you choose; you are paid the proportion of heads so far. Stopping after a first head pays one, so the only question is what to do after a tail. Waiting for the first moment heads lead pays exactly π/4 on average. Waiting for a lead of two pays exactly ln 2. The best possible rule pays about 0.79295, a number known to six decimal places only since 2013 — and it stops when the lead reaches about 0.84 times the square root of the number of tosses, a constant that comes from Brownian motion.
No set fills half of every interval
A fat Cantor set has length one half and contains no interval, so every interval holds some of it and some of its gaps. Look closely at any one point, though, and the window around it ends up almost all set or almost all gap. Lebesgue proved in 1910 that this is forced for every set with a length: the share of a shrinking window is 0 or 1 almost everywhere. So no set can fill exactly half of every interval, however it is built.
Named alongside it
The objects these essays reach for when they reach for this one.
Random walkExpectationMarkov chainBayes factorBeta distributionConditional probabilityFinite automatonLikelihoodP-valueRecurrenceStopping timeAbsorbing state