Tree Diagrams and the Independence Case

Lesson 2 of 2: Multiply Along, Add Across

In this lesson:

• Organize sequences with tree diagrams
• See independence as the special case

Advanced (+) standard — STEM-track scope.

What You Will Be Able to Do

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

1. Build a tree diagram for a two-stage dependent experiment
2. Multiply along a path and add across paths
3. Recognize independence and interpret a joint probability

One of Each Color: What's the Chance?

An urn holds 3 red and 2 blue marbles. Draw two, without replacement.

What is ?

There's more than one way to get one of each. How would you keep track?

Recap the Rule as a First Branch

The Multiplication Rule said: .

A tree draws this:

• First branches = the first draw
• Later branches carry the conditional second probability

The Tree and Its First-Level Branches

First draw: red is 3/5, blue is 2/5.

The Second-Level Branches Are Conditional

After the first draw, the urn has changed:

• After red: 2 red, 2 blue left → ,
• After blue: 3 red, 1 blue left → ,

Multiply Along a Single Tree Path

Each complete path is a sequence of "and" steps:

Along a path, you multiply — it's the Multiplication Rule on the branches.

Add Across Paths for "One of Each"

"One of each" happens two ways — add their path probabilities:

P(one of each) = 3/5

Finish the addition:

Multiply along each path, then add the two paths — one of each color is 3/5.

Multiply Along, Add Across the Tree

Two moves, two directions on the tree:

• Along a path = "and" (sequential) → multiply
• Across paths = "or" (different ways) → add

Moving along a path multiplies; combining whole paths adds.

The Four Paths Sum to 1

A built-in check: every complete path, added up, must total 1.

If your paths don't sum to 1, a branch probability is wrong.

Multiply or Add for This Step?

You want on the tree.

Do you multiply the two branch probabilities, or add them?

Decide before advancing — which direction are you moving?

When the First Doesn't Change the Second

What if drawing with replacement — returning the marble each time?

Then the urn resets, and the second draw's probabilities don't change.

That's independence — and the tree simplifies.

Independence: P(B | A) = P(B)

When and are independent, knowing tells you nothing about :

• The conditional factor becomes the plain probability
• The simple product rule is this special case

Compare: With vs Without Replacement

• With replacement (independent):
• Without (dependent):

Why the Without-Replacement Answer Is Smaller

Removing the first king makes a second king less likely:

Fewer kings remain, so the conditional second factor (3/51) is smaller than 4/52.

Your Turn: Compute and Interpret

A bin has parts, and you sample two without replacement. Given and :

Find , then write what it means in context.

Compute, then say it in plain words about the parts.

Two Common Errors to Watch For

Adding along a path: like instead of multiplying
Along a path is "and" — multiply. Add only across separate paths.

Wrong conditional: mixing with
With a sequence, condition the later event on the earlier one.

Multiply Along, Add Across — Then Interpret

✓ Later tree branches carry conditional probabilities
✓ Multiply along a path; add across paths
✓ Independent events reduce the rule to

Next: permutations and combinations — counting the huge favorable and total sets these probabilities need.