COSMO · STUDY + TEACHING DECK

Probability

February 15, 2026

A question worth sharing

Bayes' Theorem

Reverse a conditional probability by combining evidence with prior prevalence.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use prior, likelihood, posterior 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 test detects 99 percent of cases and falsely flags 1 percent of non-cases. Why can most positives be false when the condition is very rare?

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.

prior

probability before evidence

likelihood

probability of evidence under a hypothesis

posterior

updated probability after evidence

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

The core idea

Keep the idea. Explain the conditions.

\Pr(H\mid E)=\frac{\Pr(E\mid H)\Pr(H)}{\Pr(E)},\qquad\Pr(E)>0

The idea

Reverse a conditional probability by combining evidence with prior prevalence.

What to remember

Evidence must be interpreted together with how common each hypothesis was beforehand.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

A condition affects 1 percent, sensitivity is 90 percent, and false-positive rate is 5 percent. Find P(condition|positive).

Predict each next step

True positives occur with probability 0.01(0.90)=0.009.
False positives occur with probability 0.99(0.05)=0.0495.
All positives total 0.0585.
The posterior is 0. $\frac{009}{0}$ .0585, about 15.4 percent.

Which step needs a reason, not just a calculation?

See the structure

The structure behind bayes' theorem

\Pr(H\mid E)=\frac{\Pr(E\mid H)\Pr(H)}{\Pr(E)},\qquad\Pr(E)>0

A likelihood becomes a posterior only after combining it with the prior and normalizing.

Read the representation

Reverse a conditional probability by combining evidence with prior prevalence.

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 bayes' theorem

Your shared task

Students build frequency tables with 1000 cases before writing Bayes' formula.

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

Evidence must be interpreted together with how common each hypothesis was beforehand.
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

A bag is chosen equally: bag A has 2 red, 1 blue; bag B has 1 red, 2 blue. Given red, find P(A).

Predict each next step

P(red|A)= $\frac{2}{3}$ and P(red|B)= $\frac{1}{3}$ .
Multiply by equal priors $\frac{1}{2}$ .
Normalize weights $\frac{1}{3}$ and $\frac{1}{6}$ .
P(A|red)= $\frac{2}{3}$ .

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Posterior odds equal prior odds times the likelihood ratio.

Build the argument

Write Bayes for the hypothesis and its complement.
Divide the two posterior probabilities.
The shared evidence denominator cancels, leaving prior odds times likelihood ratio.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

A spam filter flags 80 percent of spam and 10 percent of legitimate mail. If 20 percent of mail is spam, find P(spam|flagged).

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

Bayesian updating appears in testing, classification, forecasting, and evidence-based reasoning.

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 test detects 99 percent of cases and falsely flags 1 percent of non-cases. Why can most positives be false when the condition is very rare?

Question 2

Explain why this claim is true: Posterior odds equal prior odds times the likelihood ratio.

Question 3

A spam filter flags 80 percent of spam and 10 percent of legitimate mail. If 20 percent of mail is spam, find P(spam|flagged).

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

Evidence must be interpreted together with how common each hypothesis was beforehand.

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

A good question belongs to everyone in the room.