Conditional Probability Across Representations

Lesson 2 of 2: One Move, Many Pictures

In this lesson:

• Find from lists, Venns, and tables
• Interpret every answer in the model

What You Will Be Able to Do

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

1. Find from a list, a Venn diagram, and a two-way table
2. Confirm the same answer appears across representations
3. Interpret each conditional probability in the model's context

The Same Probability in Three Pictures

A survey records students by sport played and grade level.

The same conditional probability can be drawn three ways:

• as a list of students
• as a Venn diagram
• as a two-way table

Is it really the same move each time? Let's find out.

Recall: The Counting Fraction Is One Move

From last lesson:

• Restrict to B — B is the new sample space
• Count A among B's outcomes
• Form the fraction

From a List: Cross Out, Then Count

Sample space, conditioning on :

• Write every outcome
• Cross out every outcome not in B
• Count how many survivors are in A

From a Venn: Overlap Over B

The denominator is the whole B circle — not the whole rectangle.

From a Table: Joint Cell Over B

• Restrict to B = use B's row (or column) total as the denominator
• Numerator is the A-and-B cell

One Move, Many Pictures — All Agree

In a Venn, What's the Denominator?

You want from a Venn diagram.

Which region is the denominator?

• A. The whole rectangle (entire sample space)
• B. The whole B circle

Pick one before advancing — one of these is the classic trap.

Now Say What the Number Means

You can compute from any picture.

The standard asks for one more thing: interpret it.

A bare fraction isn't finished — it needs a sentence about the actual situation.

Interpret: "Among the B's, This Fraction Are A"

The interpretation template:

Among the B's, this fraction are A.

• "Among the face cards, one in three is a king."

Flip the Conditioning, Change the Answer

Same two events, opposite conditioning:

"One in three face cards is a king" vs. "every king is a face card."

They Differ Because the Reference Group Changes

and are not the same.

• divides by 12 face cards
• divides by 4 kings

Same overlap (4); different denominators — so different answers.

Your Turn: Compute and Interpret

A club has 8 tenth-graders; 3 of them are officers.

Find — then write its interpretation sentence.

Compute the fraction, then say it: "among the ___, this fraction are ___."

Two Common Errors to Watch For

Swapping directions: treating as equal to
Different reference groups — check which event is after the bar.

Stopping at the number: reporting a fraction with no meaning
Always finish: "among the B's, this fraction are A."

One Move, Many Pictures — Always Interpret

✓ Restrict to B in any picture: survivors, B circle, B's row
✓ The numerator is always the A-and-B overlap
✓ Finish with a sentence: "among the B's, this fraction are A"

Next: combining events — the Addition Rule for "or" and the Multiplication Rule for "and."