Conditional Probability and Independence

Lesson 2 of 2: Restricting and Deciding

In this lesson:

• Estimate conditional probabilities by restricting to a row
• Decide independence and separate association from causation

Goals for This Second Lesson

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

1. Estimate a conditional probability by restricting to a row or column
2. Recognize the result is an estimate from a sample
3. Decide independence from the table
4. Separate association from causation in interpretation

Only Tenth Graders — Does Science Change?

Last lesson, used all 120 students.

• Now ask only about tenth graders
• Among just them, is science more or less likely?

You can't use the grand total — you must restrict. That's conditioning.

Gray Out Every Other Row

To condition on tenth grade, restrict to that row only:

Only the 60 tenth graders exist now — the restricted row is the sample space.

The Conditional Denominator Is a Row Total

For a conditional probability, the denominator changes:

• 's row or column total — not the grand total
• The move students miss most

Conditioning restricts the sample space to the given group.

Compute P(Science Given Tenth Grade)

Restrict to the tenth-grade row (60 students); 24 favor science.

• Denominator is 60 (the tenth-grade total), not 120
• Higher than the overall

The restricted denominator makes it conditional.

The Same Numerator With Two Denominators

Same 24 on top — the denominator decides which probability you get.

Compare Conditional Probabilities Across Grades

• Tenth graders favor science more than eleventh graders
• Restrict to each row, then compare

Comparing across groups is the standard's central task.

These Are Estimates From a Sample

The probabilities you compute here are approximate:

• A different sample would give slightly different numbers
• Larger samples give better estimates

Say "approximately 0.40" — these are estimates, not exact values.

Quick Check on the Right Denominator

What is the denominator for ?

Answer before advancing — the given clause sets it.

Answer: the eleventh-grade row total (60), not the grand total.

Comparing Conditional to Marginal Decides Independence

The comparison you just made is the independence test:

• → independent
• → associated

Comparing the conditional to the marginal decides it.

Decide That They Are Associated

• The two differ, so science and grade are associated
• Tenth graders favor science more than the overall rate

They're related in this sample — not independent.

Equivalent Check: Joint Versus Product

A second route to the same decision:

• — joint exceeds the product → associated
• Same conclusion as the conditional-vs-marginal test

Two routes, one answer.

Association Is Never the Same as Causation

An association does not mean one variable causes the other:

• Many explanations: scheduling, sampling, a hidden factor
• Association is symmetric; causation has a direction

Related in this sample ≠ grade causes the preference.

Your Turn: Decide and Critique

1. Is favoring English independent of grade? Compute and compare.
2. A student says "tenth grade makes people like science." Critique it.

Work both yourself. A difference shows association — not cause.

Watch Out: Two Errors to Retire

Restricted denominator: conditional divides by the given row/column total, not the grand total.

Association ≠ causation: a difference means related, not caused.

Read the "given" clause first; interpret cautiously.

Key Takeaways and What's Next

✓ Conditional: restrict to a row/column; divide within it
✓ Decide independence: conditional vs marginal (or joint vs product)
✓ Results are estimates from a sample

Association in a sample is not causation

Next: computing probabilities exactly, and the Addition and Multiplication Rules.