Compound Event Probability

Where Did Probability Come From?

Have you ever wondered why we can predict how likely something is to happen? People didn't always have a way to do that. For centuries, games of chance—like rolling dice or drawing cards—were just mysterious fun. It took curious thinkers to realize that math could explain these situations. Here's a quick look at how the idea of probability grew over time.

The big question all of these thinkers tried to answer was: When an event can happen in more than one step—like flipping two coins—how do you figure out the chance that a specific result happens? That's the question of compound event probability, and that's exactly what you'll learn here.

Core Definitions & Principles

Before we dive into compound events, let's make sure we're on the same page with a few important vocabulary words. Think of these as your building blocks.

Seeing the Sample Space

Let's look at one of the most common compound events: flipping a coin and rolling a die at the same time. A coin has 2 outcomes (Heads or Tails) and a die has 6 outcomes (1 through 6). When we combine them, the total sample space has 2 × 6 = 12 outcomes. The diagram below shows every single one.

Look at the tree diagram above. It starts at a single point, then branches into two paths—one for Heads and one for Tails. Each of those branches splits into six more paths, one for each number on the die. At the very end, you can read off every possible outcome, like (H, 3) or (T, 5). Since there are 12 endings, the sample space contains 12 outcomes.

Now, suppose you want to find the probability of getting Heads and an even number. Just count the outcomes that match: (H, 2), (H, 4), and (H, 6). That's 3 favorable outcomes out of 12 total. So the probability is 3/12, which simplifies to 1/4.

The Math Behind It

Let's write down the formula you'll use over and over again. It works for any compound event, as long as all outcomes are equally likely.

Here are two important sub-ideas that help you count the outcomes correctly.

This rule is called the Counting Principle (sometimes called the Fundamental Counting Principle). It tells you how many total outcomes are in the sample space without listing every single one. You just multiply the number of choices at each step.

Every time you solve a compound probability problem, you'll follow these three steps: Step 1 — List or count all outcomes in the sample space. Step 2 — Identify the favorable outcomes (the ones that match your event). Step 3 — Divide favorable by total, and simplify the fraction if you can.

Organized Lists & Tables

Tree diagrams aren't the only way to map a sample space. Another popular method is an organized table (sometimes called a grid or matrix). Let's use the same example—flipping a coin and rolling a die—to see how a table works.

See how the table lays out every outcome in a neat grid? The rows represent one event (Heads or Tails), and the columns represent the other event (the die numbers 1–6). Every cell is one outcome, and you can count them all: 12 outcomes total.

Now let's look at a second diagram. Imagine you are choosing an outfit by picking one shirt color and one pants color. You have 3 shirt colors (Red, Blue, Green) and 2 pants colors (Black, White). Let's build the sample space as a visual grid.

In the grid above, the two highlighted cells are the favorable outcomes for the event "picking a blue shirt." There are 6 outcomes total (3 shirts × 2 pants = 6). So the probability is 2/6, which simplifies to 1/3.

Worked Example

Let's solve a complete problem from start to finish. Follow along carefully—this is the pattern you'll use on homework and tests.

Simple Events vs. Compound Events

You might be wondering: how is a compound event different from a simple event? Let's compare them side by side so you can see clearly.

Notice that the formula for probability is identical in both cases—favorable outcomes divided by total outcomes. The only difference is that compound events have a bigger sample space, so you need better tools (like tree diagrams or tables) to keep track of everything.

Where This Leads Next

What you've learned in this lesson is the foundation for much more advanced probability work. As you continue in math, here's what builds on top of these ideas.

In high school, you'll learn about independent events (where one event doesn't affect the other, like flipping two coins) and dependent events (where one event changes the other, like drawing two cards from a deck without replacing the first). You'll also meet the Multiplication Rule, which is a shortcut so you don't have to list every single outcome. But all of that starts right here, with the skill of counting outcomes and writing fractions. Master this, and the rest will feel natural!

Practice Problems

Try these five problems on your own. Use a tree diagram, a table, or just the Counting Principle—whatever works best for you. Click "Show Answer" when you're ready to check your work.

Lesson Summary