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 - Tree Diagram

A new restaurant has opened and they offer lunch combos for $5.00.  With the combo meal you get 1 sandwich, 1 side and 1 drink.  The choices are below.

Sandwiches:  Chicken Salad, Turkey, Grilled Cheese

Sides:  Chips, French Fries, Fruit Cup

Drinks:  Soda, Water

Draw a tree diagram to find the total number of possible outcomes.

Example of a tree diagram

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 -Using the Fundamental Counting Principle

Fundamental Counting Principle

If you have a ways of doing event 1, b ways of doing event 2, and c ways of  event 3, then you can find the total number of outcomes by multiplying:

a x b x c

This principle is difficult to explain in words.  To find the total number of outcomes for the scenario, multiply the total outcomes for each individual event.

For Example 1:

3 choices of sandwiches • 3 choices of sides • 2 choices of drinks


3 • 3 • 2 = 18 total outcomes

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

The Bagel Factory offers 12 different kinds of bagels and 4 types of cream cheese.  How many possible combinations of bagels and cream cheese are there?



12 kinds of bagels • 4 types of cream cheese = total outcomes

12 • 4 = 48


There are 48 different combinations of bagels and cream cheese.


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

A pair of dice is rolled once.  How many possible outcomes are there?   What is the probability of rolling doubles?



  • Use the fundamental counting principle to find the total outcomes:

6 sides on die 1 • 6 sides on die 2 = total outcomes

6 • 6 = 36

There are 36 total outcomes.


  • Finding the probability of rolling doubles:

        There are 6 sets of doubles (1,1:  2,2:  3,3:  4,4:  5,5:  6,6)

        6   =   1        6 chances of rolling doubles

        36      6        36 total outcomes


The probability of rolling doubles is 1/6, or .167 or 16.7% chance.

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 to probability of independent events.

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