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Concept

Probability

A number between nought and one measuring how much of a space of outcomes an event takes up, adding over disjoint events. It is a measure normalised to total one, and everything about it follows from that.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The share of arrangements that fix nothing, up to 8 objects. A bar per number of objects, giving the proportion of its arrangements that leave nothing in place, against the horizontal line at 1/e.

The constant that counts what does not happen

Nothing grows in a shuffled pack of cards, and nothing grows in a factorial. Yet e sits in the middle of both — as the chance that a shuffle leaves nothing in place, and as the base that makes n! nearly a power.

analysis · The exponential A game that stops, over totals 0 to 5. States in a row with arrows up and down between them and the two ends absorbing, above a table of the expected number of steps and the chance of ending at the top from each start.

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 · Markov chains A walk on a weighted graph, and a cycle whose traffic goes one way. A weighted graph with the long-run share of each state read off its total weight, beside a three-state cycle whose shares are equal and whose traffic circulates.

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 · Markov chains The time a single walk spends in each state, against the share it should hold. Paired bars for each state, one the fraction of a long run's time spent there and one the computed stationary share, above a table of expected return times.

The time spent and the share held

The first rung's shares were a limit of distributions — where the walk probably is after many steps. This one 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 · Markov chains One walk on the whole numbers, three chances, three different fates. The relative weight of each state for three step-up chances, drawn as bars, with the running total of those weights and what each case means beneath.

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.

probability · Markov chains

Named alongside it

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

Markov chainRandom walkExpectationInvariantLimitStationary distributionAbsorbing stateApproximationCounting argumentDerangementDetailed balancee, the number

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