The Distribution the Rule Leaves Alone
Last timeWhat Is Reachable, and What Traps You
One spread of chances across the situations comes out of a hop exactly as it went in. Find it by balancing what flows into each situation against what flows out, and the long run is settled.
The second lesson showed that the distribution after many hops is the table
applied many times, and that for most chains the rows of the high powers become
identical. The third sorted out which chains those are.
This lesson asks what that common answer is, and finds it without computing any
powers at all. The trick is to stop asking what the chain tends towards and ask
instead what a single hop leaves unchanged. That question is algebra.
The condition
Take a spread of chances across the situations, adding to one, and apply the
rule once. Something comes out. A resting distribution is one where what comes
out is what went in.
The same thing as bookkeeping
Written one situation at a time, the condition says something plainer. Chance
leaves a situation at a rate equal to its share multiplied by the chance of
hopping away. Chance arrives from everywhere else. Resting means the two are
equal, everywhere, simultaneously.
The lesson stops here
6 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents