RecallWeave
Optional course · 14 questions
Three-state systems · works offline

Markov chains

Where will the probability go?

Give each state a rule. Follow the mass one step at a time, then compare a settling distribution with one that keeps alternating.

These are hypothetical teaching systems. The trace computes probabilities for each step; it does not simulate a single path or make a real-world forecast.

1. Choose the rules

Transition percentages

Each row starts in one state. Columns say where it goes next. Use whole percentages; each row must total 100%.

CurrentTo ATo BTo C
From A
From B
From C

Initial distribution

Where is the probability at step 0? These three entries must also total 100%.

Applied system: Mixing

2. Follow the distribution

State A—
State B—
State C—

Total probability: —. Displayed percentages are rounded to four decimal places.

All 30 transitions are calculated.

Before the first transition

Step 0 is the distribution you supplied. Move to step 1 or later to see the incoming contributions.

See all 31 distributions
Step 0 and 30 transitions. Percentages are rounded to four decimal places; a displayed 0% can be a small positive value.
StepState AState BState C

3. Keep an observation or learn the course

The observation file records the applied transition table, initial distribution, selected step and all computed values. The course is a separate, original fourteen-question lesson for RecallWeave.

Open RecallWeave, choose the downloaded course under Bring your own lesson, inspect its preview, then select Start this deck. The lesson opens in that page; this explorer stays a separate activity. When using a copied explorer by itself, open your existing RecallWeave file separately.

Try, explain, compare

A rule for each state, a distribution for the whole system

A transition row is conditional: it answers where the system goes next given its current state. A distribution says how likely the current states are. To find the next distribution, multiply each row by that state's current probability, then add the incoming contributions by column.

  1. Trace one calculation. Choose Mixing and move to step 2. The previous distribution is 60%, 20%, 20%. The contributions to A are 36%, 8%, 8%, which total 52%. A single A → B → C path has probability 2%; being in C after two steps includes other paths and has probability 24%.
  2. Separate stationarity from a motionless path. Choose Alternating. All in A switches between 100/0/0 and 0/50/50. Apply 50/25/25: the distribution is now stationary, even though the transition matrix makes every individual path change state at each step.
  3. Ask what the initial distribution changes. Choose Two closed states. All in C reaches 40/60/0 after one step; All in A stays in A. The same matrix can have more than one stationary distribution.

For a finite irreducible chain, a stationary distribution is unique. Adding aperiodicity guarantees convergence to it from every initial distribution. Irreducibility alone does not guarantee convergence, as the Alternating example shows. These are sufficient conditions; a reducible chain can also have a unique stationary distribution, as Absorbing A does.

The editable systems use three states, fixed transition rules, whole percentages and a 30-step horizon. Computation uses JavaScript binary64 arithmetic. Neither a finite plot nor rounded labels prove an exact limiting result. An absorbing state has a 100% self-transition; that fact alone does not show that other states can reach it.

Mathematical references and teaching notes

The questions, numbers and explanations are original. The following primary lecture notes support the standard definitions and theorems. These optional external links need an internet connection; the explorer and course do not.

Course guide and worked checks accompanies this page in the source repository.