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Compound Events and Counting Methods

Students model multistep chance processes and calculate probabilities using organized lists, trees, tables, and counting rules.

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Beginner 15–20 minutes Probability & Statistics
Probability & Statistics Page 6 of 8
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At a Glance

Compound Events and Counting Methods
Binder SectionProbability & Statistics
DifficultyBeginner
Estimated Time15–20 minutes
PrerequisitesNone
Essential Question

How can a sample space reveal probabilities of compound events?

Build Understanding

A compound event combines two or more events. Use tree diagrams, tables, or organized lists to build the sample space. The multiplication principle counts a sequence of choices.

For independent events, multiply probabilities. For mutually exclusive events, add probabilities. When events overlap, subtract the overlap once. Permutations emphasize order; combinations do not.

Visual Model
4Thousands 3Hundreds 2Tens 7Ones

4,327 = 4,000 + 300 + 20 + 7

Goals

Learning Targets

  • Use trees or tables to organize outcomes.
  • Independent: P(A and B) = P(A)P(B).
  • Mutually exclusive: add probabilities.
  • Subtract overlap when events can occur together.
  • Order matters in permutations, not combinations.
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Interactive Vocabulary

Reveal one step at a time

Worked Examples

Example 1

Flip a coin and roll a die

1

12 equally likely ordered outcomes.

Example 2

Draw two red counters without replacement

1

The second probability changes, so events are dependent.

Ask and explain

Common Questions

Are mutually exclusive events independent? Usually no.
When do I multiply? For successive independent choices or the counting principle.
When does order matter? When different arrangements count as different outcomes.
Notice and correct

Common Misconceptions

Adding probabilities for an “and” event

For independent stages, multiply.

Replacing after a draw without stating it

Replacement determines independence.

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Compound Probability lesson and examples

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Quick Start

Warm-Up

Warm-up 1

Directions: Solve the problem. Show your thinking, then enter only your final answer.

Flip two fair coins. Find P(two heads).

Picture the coinsUse the picture to organize what you know before you solve.
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You Do

Independent Practice

Independent 1

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Flip two fair coins. Find P(two heads).

Picture the coinsUse the picture to organize what you know before you solve.
25¢10¢5¢1¢

Choose how you want to work or answer.

Independent 2

Directions: Solve independently. Show your work, check it, then enter only your final answer.

Create and solve a second example with different values.

Picture what you are countingUse the picture to organize what you know before you solve.
12345678

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4
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Challenge & Real-World Practice

🧩 Challenge

Build a multistage game with a target probability between 0.20 and 0.30. Prove the exact probability and evaluate fairness.

IXL

Khan Academy

DeltaMath

DeltaMath

DeltaMath: Counting Principle, Permutations, and Combinations

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Question 1

What should happen before calculating?

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Question 2

Which response gives the strongest evidence?

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What is an effective accuracy check?

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Why are units or labels important?

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