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Tree Diagrams

Multiply probabilities along the branches and add probabilities in columns

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Tree Diagrams

Let’s Think About It

License: CC BY-NC 3.0

After running out of the haunted house, Sheila and her friends stopped a few minutes to catch their breath. Then they each gave the door attendant another ticket and ran back inside. After 10 minutes, they exited running and screaming. When they walked back up to the entrance, Charles, who was taking tickets at the door, asked them how many tickets had they used going in the haunted house. Sheila turned and asked Tonja and Annette how many times they went in together without her. They told her they had been in 2 times before she got there. How can Sheila figure out how many tickets they have used on admission to the haunted house?

In this concept, you will learn to use the Fundamental Counting Principle and tree diagrams to calculate outcomes.

Guidance

When thinking about probability, you think about the chances or the likelihood that an event is going to occur. Calculating probability through a ratio is one way of looking at probability. You can also think about chances or probability by calculating outcomes. An outcome is an end result. When you have multiple options you can calculate an outcome or figure out how many possible outcomes there are.

There are a couple of different ways to figure outcomes, including constructing a tree diagram, which is a visual way of listing outcomes.

Let’s construct a tree diagram.

Jessica has four different favorite types of ice cream. She loves vanilla crunch, black raspberry, chocolate chip, and lemonade. She also loves two different types of cones, a plain cone and a sugar cone. Given these flavors and cones choices, how many different single scoop ice cream cones can Jessica create?

To solve this problem, you are going to create a tree diagram.

First, list the choices of ice cream.

Vanilla Crunch

Black Raspberry

Chocolate Chip

Lemonade

Next, add in the two cone types. Each flavor has two possible cone types that it could go on. This is where the tree diagram part comes in.

License: CC BY-NC 3.0

Here you have four different flavors, and two types of cones, which mean you have 8 possible ice cream cone options.

You can see that the number of choices multiplied by the number of variables gave us the total number of outcomes.

\begin{align*}4 \times 2 = 8\end{align*}

This is called the Fundamental Counting Principle (FCP) and it can be very useful if you don’t want to draw an elaborate diagram to figure out your options.

Guided Practice

Evaluate \begin{align*}C(9, 4)\end{align*}.

To figure out the possible number of combinations, simply multiply.

\begin{align*}9 \times 4 = 36\end{align*}

The answer is 36.

Examples

Find the outcomes in the following examples. You may draw a tree diagram or use the Fundamental Counting Principle (FCP) to answer each question.

Example 1

Sarah has three pairs of pants and four shirts. How many different outfits can she create with these choices?

First, multiply the number of choices by the number of variables to get the total number of outcomes.

\begin{align*}3 \times 4 = 12\end{align*}

The answer is 12 outfits.

Example 2

Travis has four different pairs of striped socks and two pairs of sneakers, one red and one blue. How many different shoe/sock combinations can Travis create?

First, multiply the number of choices (4 pairs of socks) by the number of variables (2 pairs of sneakers) to get the total number of outcomes.

\begin{align*}4 \times 2 = 8\end{align*}

The answer is 8 combinations.

Example 3

If there are 33 ice cream flavors and two types of cones, how many different single scoop ice cream cones can you create?

First, multiply the number of choices (33 flavors) by the number of variables (2 cones) to get the total number of outcomes.

\begin{align*}33 \times 2 = 66\end{align*}

The answer is 66 combinations.

Follow Up

License: CC BY-NC 3.0

Remember Cheryl, Tonja, and Annette at the haunted house?

After catching up with her friends, Tonja and Annette, at the haunted house, Sheila and her friends spent one ticket each time they went in the haunted house. Tonja and Annette went in the haunted house two times before Sheila got there. The three of them went in twice together. Sheila wants to how many tickets the three of them have spent going in the haunted house.

How can Sheila figure out how many tickets the three of them have used at the haunted house?

Sheila can draw a tree diagram or use the Fundamental Counting Principle to solve this problem.

First, using the Fundamental Counting Principle, Sheila needs to divide the problem into two.

How many tickets did Tonja and Annette use together before she came?

\begin{align*}C(2, 2)\end{align*}

How many tickets did Sheila, Tonja, and Annette use together?

\begin{align*}C(2, 3)\end{align*}

Next, figure out the possible number of combinations by multiplying.

\begin{align*}\begin{array}{rcl} 2 \times 2 = 4 \\ 2 \times 3 = 6 \end{array}\end{align*}

Then, add the two results.

\begin{align*}4 + 6 = 10\end{align*}

The answer is the three of them spent 10 tickets going in the haunted house.

Video Review

https://www.youtube.com/watch?v=wBDOCvHYckE

Explore More

Design a tree diagram or use the Fundamental Counting Principle to determine each set of outcomes.

  1. Jessica has three skirts and four sweaters. How many possible outfits can she arrange given her clothing?
  2. Kim loves ice cream. She has the option of vanilla, chocolate or strawberry ice cream and she has different toppings to put on her ice cream cone. If she has sprinkles, hot fudge and nuts to choose from, how many different ice cream cones can she create with those toppings?
  3. There are five possible surfboard designs and two possible colors. How many possible surfboards can be created from these options?
  4. Team sweatshirts come in four colors and three sizes. How many sweatshirt outcomes are possible?
  5. A diner offers six types of toast with either scrambled or fried eggs. How many breakfast options are there?
  6. The same diner is offering a special that adds orange or apple juice with the eggs and toast. How many different breakfast options are there now?
  7. If the diner adds in coffee as a beverage choice with the other options, how many different breakfast options can you have now?
  8. If the diner also adds in a choice of bacon or sausage, how many different breakfast options do you have now?
  9. An Italian restaurant offers penne pasta, shells or spaghetti with a choice of vegetable, meat or plain sauce. How many different pasta dishes are possible given these options?
  10. If they also offer a choice of Italian bread or garlic bread, how many options are possible?
  11. If they add in the choice of a Caesar salad or a tossed salad, how many meal options are there now?
  12. If they offer a choice of ice cream or cheesecake with the meal, how many meal options are there now?
  13. The Cubs have 3 games left to play this year. How many different outcomes can there be for the three games?
  14. Svetlana tosses a coin 4 times in a row. How many outcomes are there for the 4 tosses?
  15. For a new tennis racquet, Danny can choose from 8 different brands, 3 different head sizes, and 4 different grip sizes. How many different racquet choices does Danny have?
  16. Gina tosses a number cube 3 times. How many different outcomes are possible?

Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 12.17. 

Vocabulary

Addition Principle

Addition Principle

If events A and B are mutually inclusive, then P(A or B) = P(A) + P(B) – P(A and B)
Multiplication Rule

Multiplication Rule

States that for 2 events (A and B), the probability of A and B is given by: P(A and B) = P(A) x P(B).
Outcome

Outcome

An outcome of a probability experiment is one possible end result.
Probability

Probability

Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.
Tree Diagram

Tree Diagram

A tree diagram is a visual way of showing options and variables. The lines of a tree diagram look like branches on a tree.
tree diagrams

tree diagrams

Tree diagrams are a way to show the outcomes of simple probability events where each outcome is represented as a branch on a tree.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0

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