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Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Browse grades and teaching videos01 · Read and understand
What you will learn
- Read transition probabilities with a fixed convention.
- Combine all routes to a future state.
- Solve and interpret a stationary distribution.
Before you start
Conditional probabilities, multiplication, addition, and a linear equation.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
In a toy weather model, a dry day is followed by a wet day with probability 0.2, and a wet day is followed by a dry day with probability 0.3. Starting dry, find the two-day wet probability and the stationary wet probability.
Why this math matters
Some sequences depend on their current state instead of behaving like independent repeated trials. A Markov chain records how probability moves among states. Short-term predictions depend on the starting state, while a stationary distribution remains unchanged by one update. For this positive transition matrix, repeated updates converge to that stable distribution.
Set up the model
A useful answer starts with clear assumptions:
- States are dry and wet, with rows and columns ordered in that sequence.
- Rows describe the current state and columns the next state; probability vectors are rows.
- The transition probabilities stay fixed and depend only on the current state. This is not a real weather forecast.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
How can today's state shape tomorrow's probabilities?
PausedQuestion: Start with the question. Paused.
Question
Start with the question
In a toy weather model, a dry day is followed by a wet day with probability 0.2, and a wet day is followed by a dry day with probability 0.3. Starting dry, find the two-day wet probability and the stationary wet probability.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Write the transition rule
P = [[0.8, 0.2], [0.3, 0.7]]; (1, 0)P = (0.8, 0.2)
Each row sums to one. A certainly dry starting day becomes 80% dry and 20% wet after one transition.
Combine two routes to wet
P(wet after two) = 0.8(0.2) + 0.2(0.7) = 0.30
The paths dry-to-dry-to-wet and dry-to-wet-to-wet are disjoint. Add their conditional path probabilities.
Solve the unchanged wet fraction
w = (1 − w)0.2 + 0.7w ⇒ 0.5w = 0.2 ⇒ w = 0.4
At stationarity, tomorrow's wet probability equals today's. The complementary dry probability is 0.6.
The result
Starting dry, the two-step wet probability is 30%. The stationary distribution is 60% dry and 40% wet.
A stationary 40% wet probability does not make successive days independent. The next-day wet probability still changes with the current state. Convergence here is supported by the matrix's positive entries, not a property to assume for every Markov chain.
Common mistakes to catch
- Multiplying column-style probabilities into a row-style matrix changes the calculation.
- The stationary fraction describes a long-run model pattern, not a guarantee of exactly four wet days in every ten.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Starting wet, what is the two-step wet probability?
Show a hint
Combine wet-to-dry-to-wet and wet-to-wet-to-wet.
Reveal answer and explanation
0.55
0.3(0.2) + 0.7(0.7) = 0.06 + 0.49 = 0.55.
Practice 2
If dry-to-wet becomes 0.1 and wet-to-dry becomes 0.4, what is the stationary wet probability?
Show a hint
Solve w = 0.1(1 − w) + 0.6w.
Reveal answer and explanation
0.2
Rearranging gives 0.5w = 0.1, so the stationary wet fraction is 20%.
Take the idea with you
State-transition models also describe machine conditions and simple learning systems, provided the chosen states and memory assumptions are defensible.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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