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Undergraduate · Two-state Markov chains

Two-state Markov chains: State one is favored

Two-state Markov chains: investigate state one is favored with transition 1 → 2: p = 0.2; transition 2 → 1: q = 0.8.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Two-state Markov chains: State one is favored. Step: 0. State-one probability: 1. Stationary probability: 0.8. Error multiplier: 0Probabilities approach stationary balance010612probability of state 1step n → · labeled axes rescale to this model
The row-stochastic transition matrix is [[1−p,p],[q,1−q]], with both probabilities strictly between zero and one. The initial distribution is (1,0). These are exact distribution updates, not simulated individual trajectories.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Step
0
State-one probability
1
Stationary probability
0.8
Error multiplier
0

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From experiment to screen.

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Understand what you are seeing

The idea behind the motion.

A distribution can settle into a stationary balance while individuals continue changing state. Starting entirely in state one, the distance from stationarity is multiplied by 1−p−q each step. A negative multiplier causes alternating approaches without making any probability negative. This investigation starts with Transition 1 → 2: p = 0.2; Transition 2 → 1: q = 0.8. Predict the result before playing, then change one parameter while keeping the other fixed to test the reason for the change.

A relationship to keep

π₁=q/(p+q); Pₙ(1)=π₁+(1−π₁)(1−p−q)ⁿ

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Set up the mathematical model

    Read p as the probability of leaving state one and q as the probability of returning from state two. The starting case is “State one is favored.”

  2. STEP 2

    Follow the changing quantity

    Update the state distribution twelve times and compare its first component with the stationary horizontal line.

  3. STEP 3

    Explain and test the result

    Check the balance π₁p=(1−π₁)q. Compare monotone, one-step, and alternating convergence using the sign of 1−p−q.

Your turn to explain

Make a prediction. Test your reasoning.

Keep Transition 1 → 2: p = 0.2; Transition 2 → 1: q = 0.8. Pause the timeline at 100%. Given step = 12, calculate state-one probability, stationary probability, error multiplier. Show the substitution into the displayed formula.

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

Balance gives π₁=0.8/(0.2+0.8)=0.8. Starting in state one, Pₙ(1)=π₁+(1−π₁)(1−0.2−0.8)^12=0.8. Results: State-one probability: 0.8; Stationary probability: 0.8; Error multiplier: 0. Decimal values are rounded; retain the original parameters when checking.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Undergraduate · Statistics

Verify stationarity through balanced transitions

Solve and check a stationary distribution of a two-state Markov chain.