Compound Events and Sample Spaces

Lesson 1 of 2

In this lesson:

• Define compound events and combined sample spaces
• Build organized lists and tables for two-stage experiments
• Construct tree diagrams for multi-stage experiments

Lesson 1 Learning Objectives Today

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

1. Define a compound event as an event involving two or more simple events
2. Construct a sample space using organized lists, tables, or tree diagrams
3. Identify outcomes in the sample space that satisfy a compound event

How Many Outcomes Do You Expect?

You flip a coin and roll a die at the same time.

• Coin outcomes: H, T (2 outcomes)
• Die outcomes: 1, 2, 3, 4, 5, 6 (6 outcomes)

Predict: How many combined outcomes are there?

Simple Events vs Compound Events

A simple event involves a single stage:

• Rolling one die → outcomes: {1, 2, 3, 4, 5, 6}

A compound event involves two or more stages:

• Roll a die AND flip a coin → outcomes are pairs
• Draw two cards → two stages, order matters

Counting Outcomes: The Multiplication Principle

For independent stages, combined sample space size equals the product of each stage's count.

Building the Organized List: Coin + Die

List (stage 1, stage 2) — coin first, die second.

12 outcomes ✓

Check-In: Sample Space for Two Coins

You flip two coins — call them Coin 1 and Coin 2.

1. How many total outcomes should there be? Show your multiplication.
2. List all outcomes using the (Coin 1, Coin 2) convention.

Two-Coins Answer: Four Ordered Outcomes

outcomes ✓

{HH, HT, TH, TT} — HT ≠ TH

Tables Handle Larger Two-Stage Spaces

For two dice: each die has 6 outcomes → $6 \times 6 = $ 36 outcomes total.

Listing 36 pairs in a row is error-prone — a table is faster and complete.

Two-Dice Table: All 36 Pairs

All 36 pairs organized in one grid.

Using the Table to Find Doubles

• Doubles = pairs where both dice show the same number
• Favorable outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
• Count: 6 favorable out of 36 total

Check-In: Find Sum = 7 in the Table

Using the two-dice table:

How many outcomes have a sum of 7?

List the pairs, then write the probability.

Check-In Answer: Sum = 7

Pairs that sum to 7:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

Count: 6 favorable outcomes

When Tables Fall Short: 3+ Stages

Tables work for exactly 2 stages. For 3 or more stages, use a tree diagram.

Example: flip a coin three times.

• Each flip: 2 outcomes (H or T)
• Total: outcomes
• A 2-row table can't show this — we need branches

Tree Diagram: Three Coin Flips

8 paths from root to leaf = 8 total outcomes

Reading the Tree: Finding Outcomes

From the 3-coin tree, the 8 paths give us:

Sample space: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Event: "at least 2 heads" — outcomes with H appearing 2 or 3 times:
→ {HHH, HHT, HTH, THH} — 4 paths

Building a Tree: Spinner and Coin

Spinner has 3 sections (R, B, G); coin has 2 outcomes (H, T) → paths.

Draw the tree: 3 branches for the spinner, then split each into H and T.

Verify: 6 leaves at the bottom.

Choosing the Right Sample Space Tool

Always verify: total outcomes = product of stage counts.

Lesson 1 Key Takeaways and Warnings

• Compound event = 2+ stages; outcomes are combinations across stages
• Total outcomes = — verify before computing probability
• Tools: list, table (2 stages), tree (3+ stages)
• Label in stage order; count leaves in trees

Next Up: Probabilities and Simulation

Lesson 2 covers:

• Compute from the sample space using favorable/total
• Design and run simulations for complex compound events

The sample spaces you built today are the foundation.