Analyzing Decisions With Probability

Lesson 1 of 2: The Frame and Testing Decisions

In this lesson:

• Use a general frame to analyze any decision
• Compute from a two-way table
• See how the base rate drives the answer

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Analyze a decision: strategies, outcomes, probabilities, payoffs
2. Build a two-way table and compute
3. Explain the base-rate effect for a rare condition

A 99% Accurate Test Says Positive

A disease test is 99% accurate. You test positive.

What's the chance you actually have the disease? Hold a guess.

A Six-Step Frame for Decisions

Identify strategies, outcomes, probabilities, payoffs — then compare and recommend.

Three Lenses for the Comparison

The comparison step uses whichever lens fits:

• Expected value — the average outcome
• Conditional probability — testing decisions
• Risk — worst-case and error costs

Three Decisions Share One Frame

The standard names three:

• Product testing — accept or reject a batch
• Medical testing — diagnose from a result
• Pulling the goalie — a strategic gamble

Predict the Answer Before Computing

A disease test is 99% sensitive and 99% specific. The disease affects 1% of people.

Of those who test positive, what fraction are truly sick? Commit before advancing.

Two Numbers Every Test Has

• Sensitivity — the true-positive rate (catches the sick)
• Specificity — the true-negative rate (clears the healthy)
• Both are below 100%, so errors exist

Set Up a Group of 10,000

Disease affects 1% of people:

• 100 are sick
• 9{,}900 are healthy

The Sick Group: 99 True Positives

Of the 100 sick people, the 99%-sensitive test catches:

The Healthy Group: 99 False Positives

Of 9,900 healthy, 1% test positive: false positives.

Count All of the Positives

Total positive results:

The Surprising Answer Is 50%

A "99% accurate" test gives a positive person a coin-flip chance of being sick.

Why: The Base-Rate Effect Dominates

The disease is rare, so the huge healthy group's few false positives match the sick group's true positives.

Why It Matters: Confirm Before Acting

A single positive is not a diagnosis.

This is why doctors order a confirmatory test before treatment.

Your Turn: A Different Test

A disease affects 2% of people. A test is 95% sensitive and 95% specific.

Of those who test positive, about what fraction are truly sick?

Mirror: Testing a Product Batch

Product testing has the same shape:

• Producer's risk — rejecting a good batch
• Consumer's risk — accepting a bad batch

What You Learned This Lesson

✓ A six-step frame organizes any decision

✓ For testing, the lens is conditional probability

✓ The base rate can drive surprisingly low

Coming Up Next: Strategy and Risk

Next: pulling the goalie — a decision that changes two probabilities at once.

Then risk and asymmetric error costs, where expected value alone falls short.