Using the Fundamental Counting Principle to Determine the Sample Space

As we dive deeper into more complex probability problems, you may start wondering, "How can I figure out the total number of outcomes, also known as the sample space?"

We will use a formula known as the fundamental counting principle to easily determine the total outcomes for a given problem. First we are going to take a look at how the fundamental counting principle was derived, by drawing a tree diagram.

Example 1

We were able to determine the total number of possible outcomes (18) by drawing a tree diagram. However, this technique can be very time consuming.

The fundamental counting principle will allow us to take the same information and find the total outcomes using a simple calculation. Take a look.

Example 1 (continued)

As you can see, this is a much faster and more efficient way of determining the total outcomes for a situation.

Let's take a look at another example.

Example 2

I would not want to draw a tree diagram for Example 2! However, we were able to determine the total outcomes by using the fundamental counting principle.

Let's look at one more example and see how probability comes into play.

Example 3

Now it's your turn. No tree diagrams!

Practice Problem

Answer Key

Although you may think that drawing the tree diagrams is fun, it's much easier to use the formula, isn't it? I hope you had fun - now it's time to move on.