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Concept

Stopping time

A rule for when to stop a random process that decides using only what has happened so far, never what is still to come. Fair games stopped at such times stay fair on average, which turns many waiting-time questions into short calculations.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The share that provably comes down. The proportion of starting values that fall below their own start within k steps, plotted against k up to 12. The proportion rises towards one; at the largest k drawn it is 0.94.

Almost every number comes down

The Collatz conjecture is open and a great deal about it is not. Whether a number falls below its own start in the first few steps is decided entirely by its remainder on division by a power of two, and the share of numbers for which it happens can be counted exactly.

dynamics · Collatz Watching a fair coin until it looks unfair. Fair coins, 2000 runs to 100000: share reaching p<0.05 by 10³ 0.468, by 10⁵ 0.697; share reaching BF₁₀≥20 by 10³ 0.0280, by 10⁵ 0.0340.

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.

probability · Bayes

Named alongside it

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

Bayes factorBinaryCentral limit theoremCollatzCounting two waysDensityIterationLikelihoodMartingaleModular arithmeticP valueParity

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