COSMO · STUDY + TEACHING DECK

Probability

March 15, 2026

A question worth sharing

Random Walks

Repeated random steps produce patterns in position, spread, and hitting probability.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use state, step distribution, hitting time 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

After four independent fair left-right steps of length 1 from 0, what is the probability of returning to the start?

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.

state

current position

step distribution

probabilities of moves

hitting time

first time a target is reached

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

The core idea

Keep the idea. Explain the conditions.

S_n=\sum_{i=1}^{n}X_i,\qquad\mathbb E[S_n]=0,\quad\operatorname{Var}(S_n)=n

The idea

Repeated random steps produce patterns in position, spread, and hitting probability.

What to remember

Zero expected position does not mean the walker stays near zero.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Find the expected position after n fair steps from 0.

Predict each next step

Let each step be +1 or -1 equally likely.
Each step has expectation 0.
Expectations add.
The expected position remains 0.

Which step needs a reason, not just a calculation?

See the structure

The structure behind random walks

S_n=\sum_{i=1}^{n}X_i,\qquad\mathbb E[S_n]=0,\quad\operatorname{Var}(S_n)=n

Independent fair steps of +1 or -1 have zero mean but a spread of square root n.

Read the representation

Repeated random steps produce patterns in position, spread, and hitting probability.

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 random walks

Your shared task

Groups simulate walks, graph the distribution, and compare average position with average distance.

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

Zero expected position does not mean the walker stays near zero.
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 probability of position 2 after six fair steps.

Predict each next step

To end at 2, rights minus lefts is 2.
With six steps, four must be right and two left.
Choose the four right positions.
The probability is $\binom{6}{4}$ /64 = $\frac{15}{64}$ .

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A walk from 0 with independent fair unit steps has expected position 0 and standard deviation $\sqrt{n}$ after n steps.

Build the argument

Positive and negative steps cancel in expectation.
Each step has variance 1; variances add across independent steps.
The position variance is n, so standard deviation is $\sqrt{n}$ .

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

Stretch together

Take the idea somewhere new.

Transfer challenge

A walker starts at 2 and takes independent fair steps of +1 or -1 until reaching 0 or 5. What is the probability of hitting 5 first?

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

Random walks model diffusion, queue fluctuations, finance models, and randomized algorithms.

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

After four independent fair left-right steps of length 1 from 0, what is the probability of returning to the start?

Question 2

Explain why this claim is true: A walk from 0 with independent fair unit steps has expected position 0 and standard deviation $\sqrt{n}$ after n steps.

Question 3

A walker starts at 2 and takes independent fair steps of +1 or -1 until reaching 0 or 5. What is the probability of hitting 5 first?

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

Zero expected position does not mean the walker stays near zero.

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

A good question belongs to everyone in the room.