COSMO · STUDY + TEACHING DECK

Probability

January 18, 2026

A question worth sharing

Expected Value

A weighted average for decisions made before the random outcome is known.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use outcome, payoff, expected value 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 fair coin pays $10 on heads and costs$ 6 on tails. Is the game fair? What does 'fair' mean before one flip?

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

List every outcome and its probability.
Attach the net payoff, including costs.
Compute a probability-weighted average rather than an ordinary average.

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.

outcome

one possible result

payoff

the value attached to that result

expected value

sum of probability times payoff

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

The core idea

Keep the idea. Explain the conditions.

\mathbb E[X]=\sum_x x\,\Pr(X=x)

The idea

A weighted average for decisions made before the random outcome is known.

What to remember

Expectation describes the average consequence of a choice, not a guaranteed result.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

A fair die pays the face value in dollars but costs $4 to play.

Predict each next step

Average die face is 3.5.
Subtract the $4 entry cost.
Expected net value is -$0.50.
The game loses 50 cents per play on average over many plays.

Which step needs a reason, not just a calculation?

See the structure

Expectation weights outcomes by how often they occur

\mathbb E[X]=\sum_x x\,\Pr(X=x)

Weight each net payoff by its probability; fair means expected net payoff zero.

Read the representation

A rare large prize can matter less than a common small loss. Probability and payoff must be read together.

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.

Expected value

A long-run average.
Useful before repeated decisions.
Does not promise one particular outcome.

Most likely outcome

The single outcome with greatest probability.
Can differ sharply from the average value.
Does not include the size of other payoffs.

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

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

The single outcome with greatest probability.
Can differ sharply from the average value.
Does not include the size of other payoffs.

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

Audit three carnival games

Your shared task

Teams compute net expected value, identify hidden costs, and redesign one game to be fair.

Explore

6 min - outcome tables

Compare

5 min - compare games

Defend

4 min - repair one payoff

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

Costs belong inside the payoff, not outside the reasoning.
Linearity lets expectations add even when variables are dependent.
A positive average can still include substantial risk.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

A ticket wins $20 with probability 0.1 and otherwise wins$ 0. What is a fair ticket price?

Predict each next step

Multiply each payoff by its probability.
0.1 x $20 + 0.9 x$ 0 = $2.
A $2 ticket has expected net value zero.
Risk preferences can still affect whether someone wants to play.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Expectation is linear even when outcomes influence one another.

Build the argument

Write the total payoff as X + Y.
Average each possible total using probabilities.
Collect X terms and Y terms to obtain E[X] + E[Y].

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

Stretch together

Take the idea somewhere new.

Transfer challenge

A game has positive expected value but a 95% chance of losing. Describe a payoff table that could do this.

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

Insurance, repeated games, randomized algorithms, forecasting, and risk analysis all compare uncertain outcomes using weighted value.

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

Why is expected value not a prediction of the next outcome?

Question 2

Where should an entry fee appear in the calculation?

Question 3

What makes a game fair in expected-value terms?

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

Expectation describes the average consequence of a choice, not a guaranteed result.

For each worksheet game, write an outcome-probability-payoff table before calculating.

A good question belongs to everyone in the room.