01 / 21

Sample Spaces for
Compound Events

Learn to organize all possible outcomes using lists, tables, and tree diagrams: the building blocks of probability.

7th Grade Math

What Is a Sample Space?

A sample space is the set of all possible outcomes for an experiment or event. When you flip a coin, the sample space is \({H, T}\). When you roll a die, it is \({1, 2, 3, 4, 5, 6}\).

Simple vs. Compound Events

Simple Event
One action with one set of outcomes. Example: rolling a single die gives \(6\) outcomes.
Compound Event
Two or more actions combined. Example: rolling a die and flipping a coin gives \(6 \times 2 = 12\) outcomes.

Why Organize Outcomes?

With compound events, the number of outcomes can get large quickly. Without a system, you might miss outcomes or count duplicates. Three tools keep you organized:

Organized Lists
Write out every pair in a systematic order.
Tables
Row-by-column grid showing all combinations.
Tree Diagrams
Branching paths that map every sequence.

The Counting Principle

Event A
\(m\)
 outcomes × Event B
\(n\)
 outcomes =
\(m \times n\)

This principle tells you how many outcomes to expect. The three tools (lists, tables, trees) show you what they are.

Tool 1: Organized Lists

Write every outcome as an ordered pair. Fix the first event's outcome, then cycle through all outcomes of the second event before moving on.

Tool 2: Tables

Put one event's outcomes along the rows and the other along the columns. Each cell shows one combined outcome.

Coin\Die 1 2 3 4 H (H, 1) (H, 2) (H, 3) (H, 4) T (T, 1) (T, 2) (T, 3) (T, 4)
Total: \(2 \times 4 = 8\) outcomes

Tool 3: Tree Diagrams

A tree diagram branches out from each choice. Each complete path from root to leaf is one outcome.

Start H T 1 2 3 1 2 3 (H, 1) (H, 2) (H, 3) (T, 1) (T, 2) (T, 3) 6 total outcomes

Lists vs Tables vs Trees

Feature Organized List Table Tree Diagram
Best for Small sample spaces Exactly 2 events 2 or more events
Visualization Text pairs Grid of cells Branching paths
Shows order? Yes Yes Yes
Easy to count? Medium Very easy Easy
3+ events? Gets long Hard ✓ Great

Step-by-Step Approach

1
Identify the events. How many separate actions are there? What are the outcomes for each?
2
Use the Counting Principle. Multiply to predict the total: \(m \times n \times \ldots\)
3
Choose a tool. Pick a list, table, or tree diagram based on the number of events.
4
Write out all outcomes. Fill in every pair (or triple, etc.) systematically.
5
Verify the count. Does your list/table/tree match the predicted total?

Example 1: Spinner + Coin

A spinner has 3 sections (Red, Blue, Green) and you flip a coin. List all outcomes and find the probability of landing on Blue and Heads.

Example 2: Two Dice Table

Roll two dice. Use a table to show all outcomes where the sum is 7.

Die 1 \ Die 2 1 2 3 4 5 6 1 (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
6 outcomes sum to 7: \(P(\text{sum}=7) = \frac{6}{36} = \frac{1}{6}\)

Example 3: Three Events

A sandwich shop offers 2 breads (White, Wheat), 2 meats (Turkey, Ham), and 2 cheeses (Cheddar, Swiss). How many combinations? Which ones have Wheat bread?

Identifying Events in Everyday Language

Problems often describe events with words like "at least one," "both," "neither," or "exactly." You need to translate these into specific outcomes from your sample space.

"At least one head"
All outcomes except (T, T). Count everything that has \(\geq 1\) head.
"Both even"
Only outcomes where every die shows an even number.
"Neither red"
Exclude all outcomes that contain red in any position.

Common Mistakes

✕ Missing Outcomes
Jumping around randomly instead of following a systematic order. Always fix one event, then cycle through all of the other.
✕ Misunderstanding Order
Forgetting that order matters. For example, when rolling two dice, (1, 6) is a different outcome from (6, 1). Both must be counted.
✕ Adding Instead of Multiplying
Writing \(3 + 2 = 5\) instead of \(3 \times 2 = 6\). The Counting Principle always uses multiplication.

Summary of Key Ideas

Concept Key Point
Sample Space All possible outcomes for an experiment
Compound Event Two or more events combined
Counting Principle Multiply outcomes: \(m \times n\)
Organized List Best for small, 2-event sample spaces
Table Grid layout, great for exactly 2 events
Tree Diagram Branches show paths: best for 3+ events
Probability \(\frac{\text{favorable}}{\text{total outcomes}}\)

Question 1 of 5

You flip a coin and spin a spinner with 4 equal sections (A, B, C, D). How many outcomes are in the sample space?

Question 2 of 5

Which tool is best for displaying all outcomes when combining 3 different events?

Question 3 of 5

You roll two standard dice. Using the phrase "at least one die shows a 6," how many outcomes in the sample space satisfy this event?

Question 4 of 5

A bag has 3 marbles (Red, Blue, Yellow). You draw one marble, put it back, and draw again. Which outcome is NOT in the sample space?

Question 5 of 5

A restaurant offers 3 appetizers, 4 entrees, and 2 desserts. You choose one of each. What is the total number of meal combinations?