Matpic
mathematics, in pictures
Generator

A rule for moving between 3 states

A generator in the probability library, called 79 times across 17 essays. Below: what it draws with nothing chosen 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.

With nothing chosen

A rule for moving between 3 states. 3 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.

Waiting for each pattern, as the coin tilts

Waiting for each pattern, as the coin tilts. Eight curves, one per three-toss pattern, of the expected waiting time against the chance of heads on a logarithmic scale, crossing one another as the bias changes.

Conway's odds on a coin that shows heads 60 times in a hundred

Conway's odds on a coin that shows heads 60 times in a hundred. A table of the four overlap sums for two patterns on a biased coin, the odds they give, and the same chance computed from an absorbing Markov chain.

Who wins each matchup, as the coin tilts

Who wins each matchup, as the coin tilts. Curves of the second-named pattern's winning chance against the probability of heads for several matchups, each crossing or avoiding the line at one half.

The second player's best reply, as the coin tilts

The second player's best reply, as the coin tilts. A grid with a row for each first choice and a column for each chance of heads, each cell naming the best reply and its winning chance, coloured by the reply.

How much the first player can guarantee, as the coin's bias moves

How much the first player can guarantee, as the coin's bias moves. A plot of the first player's best guaranteed winning chance in Penney's game against the probability of heads, a third on a fair coin and rising past one half only when the coin is heavily biased.

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

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Probability

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.

 Probability

A walk that samples a distribution

When a distribution can be evaluated but not drawn from, a wandering point can be arranged to visit each state as often as its weight says. The rule needs no normalising constant, compares two weights and steps or stays.

 Probability

An average that never settles

The average of many independent quantities is supposed to steady as their number grows. For one famous distribution it does not steady at all — the average of a thousand draws has exactly the same distribution as a single draw, and no amount of further averaging changes it.

 Computation

Every word once, around a cycle

A cyclic string of eight bits holds all eight three-bit words, each exactly once — and the reason such a thing exists is that the constraint linking overlapping windows is itself the construction.

 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

How long until it forgets

The essays before this one settle where a chain ends up and how much time it spends there, and none of them asks how long the settling takes. That question has an exact answer, it is a single number, and it is the only thing any practical use of a chain depends on.

 Probability

The chain that runs the same backwards

Put weights on the edges of a graph, step to a neighbour in proportion to them, and the long-run share of a state is its own weight over the total — read straight off the picture, with nothing to solve. The condition that makes that work is strictly stronger than being stationary.

 Probability

The chain that stops

Give a chain a state it cannot leave and there is no long run to find — every walk ends. What is worth computing instead is how long it lasts and where it finishes, and both are exact answers to a linear system rather than limits of anything.

 Probability

The forgetting that happens all at once

A single small chain forgets its start gradually, a little more with every step. A family of large ones can do something different — stay almost perfectly informed about where it began, and then lose all of it inside a window far shorter than the wait. That cliff is the cutoff phenomenon, and it is why "seven shuffles" is an answer rather than a convention.

 Probability

The narrowest door sets the pace

How fast a chain forgets is an eigenvalue, and nobody can compute the eigenvalues of a chain worth studying. Cheeger's inequality trades the eigenvalue for a picture — the narrowest door in the state space — and pins the one between the square of the other and twice it. Both ends of that range are reached, on graphs small enough to search completely.

 Probability

The one that hardly ever comes up

Make the kinds unequally likely and the tidy decomposition into stages fails, because a stage's rate now depends on which kinds turned up rather than on how many. What replaces it is an alternating sum over every subset — and the rarest kind turns out to be nearly the whole answer.

 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.

 Probability

The time spent and the share held

Stationary shares are a limit of distributions — where the walk probably is after many steps. Here the question is about a single walk: the fraction of its time spent in each state is that state's share, and the expected wait between visits is exactly the reciprocal.

 Probability

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.

 Probability

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.

 Probability

Two patterns, one chance, different waits

HTH and HTT are equally likely in any given window of three tosses. Waiting for HTH takes ten tosses on average and waiting for HTT takes eight, and the difference is not about probability at all — it is about what a failed attempt leaves behind.

 Probability

Where the shares have nowhere to go

On finitely many states, a chain that can reach everywhere and is not forced into a rhythm settles down. Give it infinitely many and both conditions can hold while the walk leaves and never returns — or returns with certainty and takes an unbounded average time about it.

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