Independence and Medical Testing

Lesson 2 of 2: When Direction Matters Most

In this lesson:

• See independence as
• Use the base-rate method on a medical test

Goals for This Second Lesson

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

1. Express independence as
2. Show this is equivalent to the product rule
3. Build a two-way table from base rates
4. Distinguish from in context

A 95% Test, a Positive Result

A rare disease. A test that is 95% accurate. Your result: positive.

• How likely is it you actually have the disease?
• Most people guess "about 95%"

Make your guess and hold it. The real answer will surprise you.

Independence Means Conditioning Changes Nothing

Two events are independent exactly when:

• Learning happened tells you nothing new about
• If , the events are dependent

This is the most intuitive form of independence: no change.

Verify Independence on a Die

= "even," = "greater than 2." Restrict to .

Equal — so the events are independent.

Two Definitions That Say One Thing

The conditional form and the product rule are equivalent:

• Conditional form gives the meaning ("no change")
• Product rule gives a quick test

Multiply both sides of the conditional form by to see it.

Quick Check on Independent Events

and are independent, and .

What is ? Answer before advancing.

Answer: — conditioning changes nothing.

Direction Is What Makes Testing Tricky

The trap hides in the direction of conditioning:

• Accuracy is — among the sick
• You want — among the positives

Different directions — and when disease is rare, they differ a lot.

Set Up a Group of 10,000

Imagine 10,000 people; the disease affects 1%.

• 100 people are sick
• 9,900 people are healthy

Few sick, many healthy — this base rate is the key to the puzzle.

Apply the Test to Both Groups

• Sick: test positive (95% of 100)
• Healthy: test positive (3% of 9,900)
• Total positive:

The Surprising Answer: About 24%

• A positive test means only a 24% chance of disease
• Not 95% — most positives are false alarms

The test is accurate, yet a positive usually isn't disease.

Why Only 24% From a 95% Test?

The test is 95% accurate — yet a positive means only 24% disease.

In your own words, why? Commit before advancing.

Because the disease is rare, false positives from the huge healthy group outnumber the true positives.

The Two Directions Are Not Equal

• Confusing them is base-rate neglect (the prosecutor's fallacy)
• High shows association, not causation

Umbrellas and rain go together — neither causes the other.

A Factory Quality Control Example

Machine A: 60% of products, 2% defective. Machine B: 40%, 5% defective.

More products, yet less often the defect source.

Your Turn: The Spam Filter

5% of emails are spam. Filter flags spam 90%, flags good email 2%.

• An email is flagged. Find
• Use 10,000 emails — build the table, then compute

Work it fully yourself before the reveal.

Watch Out: Two Reasoning Errors

Reversed conditioning: .

Association is not causation: a high reports a pattern, not a cause.

Ask: which direction, and pattern or mechanism?

Key Takeaways and What's Next

✓ Independent ⟺ — conditioning changes nothing
✓ Build a base-rate table to find a reversed conditional
✓ A 95% test can mean only 24% when disease is rare

Watch the conditioning direction; association isn't causation

Next: everyday-language reasoning, then two-way frequency tables.