# Counting Outcomes Using Tree Diagrams

## Tree Diagrams

Often in statistics, we’re interested in knowing all the possible outcomes of an experiment. One useful way to visualize all the possible outcomes of an experiment is to create a tree diagram.

Example 1: Bob is trying to decide what to wear today. He has three different shirts (red, blue, green) and two different shoes (gym shoes, dress shoes). How many different outfits could Bob wear?

To answer this, we can draw a tree diagram.

First, we create “branches” using all the different shirts:

Then we add “branches” to represent the different shoes:

Each branch represents a different outfit that Bob could wear. For example, he could wear a red shirt with gym shoes:

Or he could wear a red shirt with dress shoes:

In total, there are six different outfits Bob could wear:

Red shirt with gym shoes
Red shirt with dress shoes
Blue shirt with gym shoes
Blue shirt with dress shoes
Green shirt with gym shoes
Green shirt with dress shoes

The formula that we use to find the total number of outcomes is:

(# of possible outcomes of first variable) * (# of possible outcomes of second variable)

In this case, there are three different shirts and two different shoes. So the total number of outfits is:

(# of possible shirts) * (# of possible shoes) = 3 * 2 = 6

Example 2: Marie is baking a cake. She has two different types of frosting (chocolate, vanilla), two different types of designs (striped, checkered), and three different types of toppings (peanuts, almonds, sprinkles) to choose from. How many different cake variations could Marie make?

To visualize this scenario, we will draw a tree diagram. First, we’ll add the branches for the two types of frosting:

Then we’ll add the branches for the two types of designs:

Lastly, we’ll add the branches for the three types of toppings:

In total, there are 12 different cake variations Marie could make.

We also could have found this answer without drawing the tree diagram by using the following formula:

Number of cake variations = (# of possible frosting flavors) * (# of possible designs) * (# of possible toppings)

Number of cake variations = 2 * 2 * 3 = 12

Example 3: Luke is building a fantasy basketball team. He needs one player for each position. The table below shows the number of players available to choose for each position.

How many different teams could Luke build?

Number of different teams = (# point guards) * (# shooting guards) * (# small forwards) * (#power forwards) *(# centers)

Number of different teams = 5 * 7 * 5 * 9 * 3 = 4,725