COSMO · STUDY + TEACHING DECK

Probability

April 26, 2026

A question worth sharing

Markov Chains

Model systems where the next state depends on the present state, not the full path.

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Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use transition, transition matrix, stationary distribution to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

A two-state chain switches states with probability 0.2 each step. What fraction of time is spent in each state long-run?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

transition

move between states

transition matrix

table of move probabilities

stationary distribution

distribution unchanged by one step

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\pi P=\pi,\qquad\sum_i\pi_i=1,\quad\pi_i\ge0

The idea

Model systems where the next state depends on the present state, not the full path.

What to remember

A transition matrix turns local movement rules into long-run behavior.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Update distribution (0.7,0.3) using matrix [[0.8,0.2],[0.4,0.6]].

Predict each next step

New first-state probability is 0.7(0.8)+0.3(0.4)=0.68.
New second-state probability is 0.7(0.2)+0.3(0.6)=0.32.
The entries sum to 1.
The updated distribution is (0.68,0.32).

Which step needs a reason, not just a calculation?

See the structure

The structure behind markov chains

\pi P=\pi,\qquad\sum_i\pi_i=1,\quad\pi_i\ge0

With row-vector distributions, a stationary distribution stays unchanged after one transition.

Read the representation

Model systems where the next state depends on the present state, not the full path.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate markov chains

Your shared task

Students act out transition rules and compare empirical state frequencies with matrix predictions.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A transition matrix turns local movement rules into long-run behavior.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Find the stationary distribution for [[0.9,0.1],[0.3,0.7]].

Predict each next step

Let stationary probabilities be (p,1-p).
Solve p=0.9p+0.3(1-p).
This gives 0.4p=0.3.
So p=0.75 and the distribution is (0.75,0.25).

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A stationary distribution satisfies pi P = pi.

Build the argument

Multiplying by P performs one transition step.
Stationarity means the distribution does not change.
Therefore the fixed-point equation is pi P = pi.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Construct a three-state chain that is not irreducible and explain why a unique stationary limit may fail.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Markov chains model weather states, web navigation, queues, genetics, and repeated decisions.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

A two-state chain switches states with probability 0.2 each step. What fraction of time is spent in each state long-run?

Question 2

Explain why this claim is true: A stationary distribution satisfies pi P = pi.

Question 3

Construct a three-state chain that is not irreducible and explain why a unique stationary limit may fail.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A transition matrix turns local movement rules into long-run behavior.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.