Independence in Plain Language and Critique

Lesson 2 of 2: Naming the Everyday Errors

In this lesson:

• Describe independence with the information test
• Catch and rewrite flawed probability claims

Goals for This Second Lesson

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

1. Describe independence with the information test
2. Tell independence apart from mutual exclusivity in words
3. Name the three everyday error families
4. Critique a flawed claim and rewrite it accurately

Red Five Times — Is Black "Due"?

Red has come up five times in a row on a roulette wheel.

• A voice says "black is due now"
• Does the wheel remember its past spins?

It feels right that things should even out. But is it?

The Information Test for Independence

For any two events, ask one question:

Does knowing the first change my expectation of the second?

• Coin flip lands heads → next flip still one-half
• No information gained → the events are independent

This single question is your tool for the whole lesson.

State Independence in Plain Words

Two events are independent when:

knowing one happened tells you nothing about the other

• No formula needed — just the information question
• Portable: apply it anywhere, instantly

If the first gives you no information about the second, they're independent.

Examples and Non-Examples of Independence

Independent: coin flips, strangers' birthdays. Not independent: umbrella and rain, study hours and grade.

Independent Is Not "Can't Co-Occur"

Independence is about information, not overlap:

• Heads and a six can happen together — yet independent
• Mutually exclusive = can't both happen (and that's dependent)

Independent = no information. Don't confuse it with "can't co-occur."

Quick Check: Two Cards, No Replacement

Two cards drawn without replacement. Are the draws independent?

Apply the information test before advancing.

Answer: not independent — the first draw changes what's left for the second.

The Test Exposes Everyday Errors

The information test catches three recurring errors:

• People reverse conditions, assume false independence, or confuse association with cause
• Naming the error is the first step to correcting it

Spot the error family, then fix the claim.

Three Error Families to Watch

Reversed conditioning, false independence, association-as-causation. Nearly every flawed claim is one of these.

Predict: Does This Reasoning Hold?

"The test is 99% accurate, so a positive almost certainly means disease."

Decide before advancing — recall last lesson's testing surprise.

The 99% is accuracy, — not .

How to Correct Reversed Conditioning

The error: reading accuracy as the chance of disease.

• Accuracy is
• You want — different for a rare disease

The fix: name the direction, and ask how rare the condition is.

False Independence: The Gambler's Fallacy

The error: thinking past independent outcomes change the next.

• "Red five times, so black is due" — the wheel has no memory
• Mirror error: assuming a "hot streak" that isn't real

The fix: independent trials have no memory — the past tells you nothing.

Association Wrongly Read as Causation

The error: treating "they go together" as "one causes the other."

• "Umbrella-carriers get rained on, so umbrellas cause rain" — absurd
• Both share a common cause (the forecast)

The fix: association reports a pattern, not a mechanism.

Your Turn: Name and Rewrite

"Most winners bought tickets at a lucky store, so buying there boosts your odds."

• Which error family is this?
• Rewrite it so it's accurate

Do both yourself. Is this reversed conditioning, false independence, or causation?

The Three Error Families Summarized

Reversed conditioning: "A given B" ≠ "B given A" — name the direction.

False independence: the gambler's fallacy — trials have no memory.

Association as causation: "goes together" ≠ "causes."

This trio covers nearly every flawed probability claim.

Key Takeaways and What's Next

✓ Independent = knowing one tells you nothing about the other
✓ It's about information, not whether events can co-occur
✓ Name the error: reversed, false independence, or causation

Critique by spotting the error family, then rewriting

Next: computing conditional probability, and the rules of probability.