Independence and the Product Rule

Lesson 1 of 2: What Independence Means

In this lesson:

• Say what it means for two events to be independent
• Test independence with the product rule

Goals for This Two-Lesson Pair

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

1. State that and are independent when
2. Explain independence as "one event doesn't change the other"
3. Test independence by computing and comparing
4. Distinguish independent events from dependent events
5. Tell independence apart from mutual exclusivity

Does the First Flip Change the Second?

You flip a fair coin and it lands heads.

• Does that change the chance the next flip is heads?
• Your gut says no — the coin has no memory

That "no effect" intuition is what we'll call independence.

Two Situations, Side by Side

Coin flips: no effect — independent. Marble draws without replacement: the pool shifts — dependent.

Independence Always Links Two Events

Independence is a relationship between two events — not a property of one.

• "A is independent" is incomplete — independent of what?
• It means: knowing happened doesn't change

Always name both events: " is independent of ."

Quick Check: Independent or Dependent?

Label each pair, justifying in one sentence:

1. Two separate dice rolls
2. Two cards drawn without replacement
3. Today's weather, tomorrow's lottery numbers

Ask: does knowing the first change the second? (1 and 3 independent; 2 dependent.)

From No Effect to a Real Test

You can describe independence in words. But how do you check it?

• "No effect" needs to become arithmetic you can verify
• The key: if doesn't change , both happening is just the product

Next: the product rule turns the idea into a test.

The Product Rule for Independence

and are independent if and only if:

• This is both the definition and the test
• Compute all three; check whether the product matches

If equals , the events are independent.

Verify It: Two Coin Flips

List the sample space: HH, HT, TH, TT — four equally likely.

The product matches — the flips are independent.

Predict: Can One Die Give Independence?

Two events on a single die: = "even," = "greater than 2."

• A. No — independence needs two separate experiments
• B. Yes — they can still be independent

Commit before advancing. The answer may surprise you.

The Die Surprise: They Are Independent

Same die, yet independent — knowing "" doesn't change "even."

Same Experiment Can Be Independent

The surprise to remember:

• Independence does not need separate experiments
• Two events from one roll can be independent

What matters is information — does knowing one shift the other? — not the number of experiments.

Your Turn: Test This Pair

On a die: = "a multiple of 3," = "greater than 3."

• Compute , ,
• Decide: independent or dependent?

Work all three yourself. Check whether before the reveal.

Watch Out: Two Beliefs to Retire

"Needs separate experiments": No — one die gave an independent pair.

"A single event is independent": No — always two events. Ask "independent of what?"

Name both events before you test.

Key Takeaways and What's Next

✓ Independent = knowing one doesn't change the other
✓ Test:
✓ Even two events on one experiment can be independent

Independence is about information, not separate experiments

Next: test independence on real survey data, and contrast with mutual exclusivity.