# Tree Diagrams, Sample Space Diagrams & Tables of Outcomes

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• 0:04 Tree Diagrams
• 1:24 Sample Space
• 1:59 Table of Outcomes
• 2:26 Example 1
• 4:01 Example 2
• 5:00 Lesson Summary
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Lesson Transcript
Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

In this lesson, we will define tree-diagrams, sample space, and tables of outcomes as related to probability. Examples of how to create tree-diagrams and tables of outcomes are also provided.

## Tree Diagrams

Have you ever opened your closet and thought to yourself, ''I have nothing to wear?'' You might be surprised to learn that if you had just three shirts and two pairs of pants, you actually have six different outfits that you could wear. Suppose you have a black shirt, a brown shirt, and a white shirt, and you also have a pair of jeans and a pair of black pants. How could we make 6 outfits?

Well, if you decided to wear the black shirt, you could choose to wear jeans or the black pants, which gives us two possible outfits. If you decided to wear the brown shirt, again you could choose to wear jeans or the black pants. Again, that gives us two possible outfits, with a total of four possible outfits. You might decide to wear the white shirt, and you could choose either jeans or black pants. These two options now give us a total of six possible outfits. We can multiply the number of shirts (3) times the number of pants (2) to get the 6 total possible outcomes.

The best way to see all these possibilities is to use a tree diagram. A tree diagram is a visual way to represent the total outcomes you could have. You can see this laid out in the diagram on your screen below.

To determine the total number of outcomes, you can just add up the number of pictures at the right of the tree diagram. We can also create a tree diagram using words instead of picture, as you can see.

## Sample Space

A tree diagram is a great way to organize the sample space of a problem. The sample space is the total number of possible outcomes. Using the example we just went through in the last section of this lesson, there are 6 items in the sample space because there are 6 total possible outcomes.

Sample space can be listed. The sample space for the outfits is {Black Shirt Black Pants, Black Shirt Jeans, Brown Shirt Black Pants, Brown Shirt Jeans, White Shirt Black Pants, White Shirt Jeans}. As you can see, there are 6 combinations listed because there are 6 total possible outcomes.

## Table of Outcomes

We can represent the sample space using a table of outcomes as well as using tree diagrams. A table of outcomes is a table where the first row and first column represent the possible outcomes in each event. For instance, the first column would be each shirt, and the first row would be each pair of pants. Then we simply fill in the table with the possible outcomes, which you can see on your screen below.

Notice that there are 6 possible outcomes listed inside the table.

## Example 1

Suppose you were going to flip a coin and roll a fair die. Draw a tree-diagram of the sample space, list the sample space, and create a table of outcomes. How many outcomes are in the sample space?

#### Step 1: Tree Diagram

The first thing we need to do is determine the two things we are doing and what the possible outcomes are. We are flipping a coin and rolling a die. We are going to assume that they are both fair, meaning that one side is not weighted more than the other. If we flip a coin, we have 2 possible outcomes: Heads or Tails. If we roll a die, we have 6 possible outcomes, 1, 2, 3, 4, 5, or 6. We are going to start with the coin, but you could start with the die.

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