Undergraduate · Two-state Markov chains · 110 of 650
Two-state Markov chains · Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.6
Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier. Givens: Transition 1 → 2: p=0.5; Transition 2 → 1: q=0.6.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A two-state chain has row transition matrix [[0.5,0.5],[0.6,0.4]] and starts entirely in state 1. Use n=7 transitions. Find the probability of state 1, its stationary probability, and the signed convergence multiplier.
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A starting point
Stationarity gives π₁=q/(p+q). The distance from it is multiplied by 1−p−q at each step.
Work through the reasoning
Step 1
Identify the model and target
A two-state chain has row transition matrix [[0.5,0.5],[0.6,0.4]] and starts entirely in state 1. Use n=7 transitions. The governing relation is π₁=q/(p+q); Pₙ(1)=π₁+(1−π₁)(1−p−q)ⁿ. Stationarity gives π₁=q/(p+q). The distance from it is multiplied by 1−p−q at each step.
Step 2
Substitute and calculate
Balance gives π₁=0.6/(0.5+0.6)=0.545455. Starting in state one, Pₙ(1)=π₁+(1−π₁)(1−0.5−0.6)^7=0.545454.
Step 3
Check the mathematical meaning
The state-1 probability 0.545454 lies in [0,1]. Stationary flows balance: π₁p=(1−π₁)q=0.272727. A negative multiplier means alternating errors, not negative probabilities. Step: 7; State-one probability: 0.545; Stationary probability: 0.545; Error multiplier: -0.1. Decimal values are rounded, so use unrounded intermediate values.
The answer
Balance gives π₁=0.6/(0.5+0.6)=0.545455. Starting in state one, Pₙ(1)=π₁+(1−π₁)(1−0.5−0.6)^7=0.545454. The state-1 probability 0.545454 lies in [0,1]. Stationary flows balance: π₁p=(1−π₁)q=0.272727. A negative multiplier means alternating errors, not negative probabilities. Animation check: Step: 7; State-one probability: 0.545; Stationary probability: 0.545; Error multiplier: -0.1. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
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Watch the relationship
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The mathematical idea
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. A two-state chain has row transition matrix [[0.5,0.5],[0.6,0.4]] and starts entirely in state 1. Use n=7 transitions. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
π₁=q/(p+q); Pₙ(1)=π₁+(1−π₁)(1−p−q)ⁿ
03 · Reflect and transfer
Explain what changes and why.
What happens when p+q=1, and why does that special case reach the stationary distribution after one update?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.