Matpic
mathematics, in pictures
Generator

chain

A generator in the probability library, called 10 times across 2 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

chain is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

A rule for moving between 3 states3 states drawn as circles with an arrow for every move the rule allows, labelled with its chance; a dashed loop is the chance of staying put..6.3.4.4.3.3.10.20.40ABCevery state can be reached from every other, and no length of walk is forcedevery arrow out of a state carries a chance, and the chances out of each state add to one

show: "collect"

Waiting for all 6 kindsOne bar per new kind: the expected number of draws needed to see a kind not yet seen, rising as fewer of them are left, and adding to 14.70 draws in total.10 → 11.201 → 21.502 → 323 → 434 → 565 → 614.70 draws expected in all6 kinds, and the last is the expensive onethe waits are 6/6 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1 = 14.70 drawsthe last one alone costs 6 draws on average, which is why the total grows faster than the number of kinds

show: "powers"

The rule applied 1, 2, 4, 8 times overThe transition matrix raised to each power in turn, drawn as shaded grids; the entry in row i and column j is the chance of being at j after that many steps having started at i.one step.10.60.30.40.20.40.30.30.40ABCABC2 steps.34.27.39.24.40.36.27.36.37ABCABC4 steps.29.34.37.27.35.37.28.35.37ABCABC8 steps.28.35.37.28.35.37.28.35.37ABCABCthe largest gap within a column falls from 0.400 to 0.000once the rows agree, the chance of being somewhere no longer depends on where the walk started

show: "stationary"

The share of the long run spent in each stateOne group of three bars per state: the share from solving the equations, the share from applying the rule 60 times, and the share of a 40,000-step walk actually spent there.A0.2791 solved0.2791 60 steps of the rule0.2791 40,000 steps walkedB0.3488 solved0.3488 60 steps of the rule0.3475 40,000 steps walkedC0.3721 solved0.3721 60 steps of the rule0.3734 40,000 steps walkedsolved exactly, reached by iterating, and walked 40,000 times — three routes to the same sharesthe walk is seeded, so it is a fixed picture rather than a fresh experiment, and it agrees to about a hundredth

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Probability

How long until every one turns up

Draw at random from six equally likely kinds until all six have appeared. The wait is not six draws, and it is not sixty; it is fourteen point seven, and the number is a harmonic sum wearing a hat.

 Probability

The rule that forgets where it came from

A walk between a few states, with the next step decided by the current one and nothing else. Run it long enough and the starting point stops mattering — but only when two conditions hold, and both of them have a picture in which they fail.

The whole library · What the figures prove