Testing Independence on Real Data

Lesson 2 of 2: Data and Mutual Exclusivity

In this lesson:

• Test independence from a two-way table
• Tell independence apart from mutual exclusivity

Goals for This Second Lesson

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

1. Test independence from a two-way table by computing three probabilities
2. Explain why real data must be computed, not guessed
3. Distinguish independence from mutual exclusivity
4. Classify event pairs and compute their joint probability

Is Good Sleep Related to Exercise?

Do people who exercise tend to sleep better?

• It feels like they should be related
• Predict: are "exercises" and "good sleep" independent?

Commit to a guess before advancing — then we compute.

Bring the Test to a Table

The test from last lesson, now on data:

• , come from the margin totals
• comes from the joint cell

The table is a sample space — every probability is a count over the total.

The Three Steps of the Table Test

1. = a margin total ÷ grand total
2. = the other margin total ÷ grand total
3. = the joint cell ÷ grand total

Then compare to . Every denominator is the grand total.

Read the Exercise and Sleep Table

Compare and Find Them Dependent

• , so the events are dependent
• Exercisers sleep well more often than independence predicts

The data settled it — not the hunch.

Quick Check: What If They Matched?

Suppose had come out to exactly 0.288.

What would that tell you? Answer before advancing.

Answer: equal to the product → the events would be independent.

Intuition Isn't Enough — Compute

With real data, the hunch can mislead you.

• Pairs that feel unrelated can turn out dependent
• Pairs that feel related can turn out independent

You can't settle it by feel — compute the three probabilities and compare.

Two Relationships People Often Blur

People confuse two different relationships:

• Independent — knowing one doesn't change the other
• Mutually exclusive — the two can't both happen

They sound alike but mean opposite things about co-occurring. Let's separate them.

Independent Events Can Happen Together

Independent events can co-occur — joint probability is the positive product.

Mutually Exclusive: Cannot Both Happen

On one die roll, you can't roll a 2 and a 5 at once.

• Mutually exclusive events have joint probability zero
• They exclude each other — nothing overlaps

Zero joint probability is the signature of mutual exclusivity.

They Cannot Both Be True at Once

If and :

• Independent → joint = positive product
• Mutually exclusive → joint =

A number can't be positive and zero — so the same pair can't be both.

Your Turn: Classify and Justify

Draw one card. Events: "it's a king," "it's a queen."

• Independent, mutually exclusive, or neither?
• Compute to justify

Work it yourself. One card can't be both — so what's the joint probability?

Watch Out: Two Errors to Retire

Independent ≠ mutually exclusive: one can co-occur (joint ); the other can't (joint ).

Don't assume independence: with real data, compute and compare.

Name the relationship; compute the joint probability.

Key Takeaways and What's Next

✓ Test independence from a table: compare to
✓ With real data, compute — don't guess
✓ Independent (can co-occur) ≠ mutually exclusive (joint )

Two positive-probability events can't be both

Next: conditional probability — independence as .