Conditional Probability and Its Formula

Lesson 1 of 2: Restricting the Sample Space

In this lesson:

• Compute by restricting the sample space
• Read conditional probability from tables, statements, and Venns

Goals for This Two-Lesson Pair

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

1. Compute
2. Explain conditioning as restricting the sample space
3. Interpret independence as
4. Test independence using the conditional criterion
5. Apply conditional probability to real-world scenarios

Recall: Sample Space and Probability

• Sample space: the set of all possible outcomes
• Probability of a simple event:
• The total in that fraction is the size of the sample space

We're about to watch that total shrink as we condition.

Not Green — Does That Change Red?

A bag has 10 marbles: 4 red, 3 blue, 3 green. You draw one.

• Plain chance of red:
• Now you learn the marble is not green

Does that change the chance it's red? You just gained information.

Recount the Smaller Marble Pool

If it's not green, only 7 marbles remain (4 red, 3 blue).

The sample space shrank from 10 to 7 — conditioning recounts in the smaller world.

Conditioning Restricts the Sample Space

Conditioning on makes the new sample space.

Zoom into ; ask what fraction of it is also . Outcomes outside don't count.

The Conditional Probability Formula Defined

• Numerator: the overlap,
• Denominator: all of , the new sample space

"What fraction of 's probability also has ?" — the zoom written in symbols.

Read It from a Two-Way Table

30 students; restrict to band (8 students), of whom 4 play sports.

Counting within the band row and the formula agree.

Quick Check: Use the Formula

Given , , .

Find before advancing.

Same Formula, Many Data Sources

The formula doesn't care where the numbers come from:

• From a table: counts within a row or column
• From a statement: given probabilities
• From a Venn: region probabilities

One move every time: find and , then divide.

Coffee and Tea: Read the Table

200 people; restrict to coffee drinkers (120), 40 of whom drink tea.

Among coffee drinkers, one in three also drinks tea.

Direction Matters: Order Is Not Symmetric

and generally differ — the denominators differ.

Same joint cell, different totals — order changes the answer.

Conditional Probability from Given Statements

Given , , .

Both exceed the plain probabilities — rain and traffic are dependent.

Conditional from a Venn Diagram

Different denominators, different answers — one formula.

Your Turn: Both Directions from a Venn

Regions: A only , overlap , B only , neither .

• Find and
• Compute and

Work all four yourself. The given event sets each denominator.

Watch Out: Two Formula Traps

? No — divide by ; it's within .

? No — different denominators, different answers.

Whatever follows the bar is the given event — and the denominator.

Key Takeaways and What's Next

✓ — zoom into
✓ Read it from tables, statements, or Venns alike
✓ in general

The bar has a direction — given event is the denominator

Next: independence and medical testing.