Conditional Probability by Counting

Lesson 1 of 2: Restrict and Count

In this lesson:

• Find by counting within B
• See why counting matches the formula you know

What You Will Be Able to Do

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

1. Compute by restricting to B and counting
2. Explain why conditioning makes B the new sample space
3. Connect the count to the formula

Given It's Greater Than 3, Is It Even?

Roll a fair die. The outcomes are .

• Let = "greater than 3"
• Let = "even"

Given that the roll is greater than 3, what's the chance it's even? Try counting — don't reach for a formula yet.

Throw Away Every Outcome Not in B

Only survive. Two of them — 4 and 6 — are even.

Conditioning Makes B the New Sample Space

When you condition on B, you replace the sample space with B.

• The roll is now one of — three outcomes
• You count A only among those three
• The denominator becomes , not the original total

Compare: 2/3 Is Not 3/6

Same event "even" — but conditioning changed the denominator from 6 to 3.

What Is the New Denominator Now?

You condition on = "greater than 3" for a fair die.

After restricting to B, how many outcomes are you counting among?

Think before advancing — what number goes on the bottom?

Write the Move as a Counting Fraction

You just did this: count A among B's survivors, divide by how many survived.

The top is the overlap — outcomes in both A and B. The bottom is all of B.

The Counting Rule for Uniform Models

In a model where every outcome is equally likely:

• — how many outcomes are in both A and B
• — how many outcomes are in B

Why It Equals the Formula You Know

Divide top and bottom by the total count :

The total cancels — counting and the formula are the same thing.

Cards: P(king | face card) = 1/3

• Among the 12 face cards (B), 4 are kings (A)

Verify the Count With the Formula

The formula must give the same answer:

The 52s cancel — exactly the cancellation from two slides ago.

The Numerator Is the Overlap Only

For , the event A = "even" is .

• But 2 is not in B — it's not greater than 3
• Count only the evens that survived:
• Numerator is , not

Your Turn: Restrict and Count

A fair die is rolled. You're told the result is odd.

What is ?

Restrict to the odds, count how many are less than 4, then form the fraction.

Two Common Errors to Watch For

Wrong denominator: dividing by the whole sample space, not by
Conditioning shrank the world to B — divide by B's count.

Wrong numerator: counting all of instead of
An A-outcome outside B (like the 2) doesn't survive — don't count it.

Restrict to B, Then Count A

✓ Conditioning makes B the new sample space
✓ — count the overlap over all of B
✓ It equals because the total cancels

Next: the same move across lists, Venn diagrams, and tables.