hover a state to trace its weather
A token wanders a weather of pure chance — and its visit-frequencies always settle onto the same long-run climate, no matter where it began.
Site 154 · Chance · a memoryless walk
The chain forgets where it started. It cannot help but arrive.
Four weather states, one rule: the next day depends only on today, never on the week behind it. That is the whole of a Markov chain — memorylessness. Yet from that thin, forgetful rule an iron regularity falls out. Tally the token's days across ten thousand hops and the fractions stop drifting. They lock onto the stationary distribution — the long-run climate that any starting weather is eventually pulled into.
Nothing coordinates the token. Each hop is a single weighted coin, read off the outgoing edges and thrown. The bar chart is just bookkeeping. What you are watching is the difference between a path — which is pure chance — and a distribution, which is law.
| from ╲ to | Clear | Cloud | Rain | Fog |
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Every row of P sums to one — a full day of certainty spent somewhere. The token stands on a state, reads that row as odds, and jumps. The thickness of each drawn arrow is its probability; the faint hairlines are the transitions that rarely happen.
The empirical bars ride up to the thin markers, which are the true π computed from P directly. When the coloured bar meets its marker, the walk has forgotten its origin. The drift readout is the total-variation distance between the two — the last trace of the starting weather, decaying toward zero.