11.2 Representing Outcomes

Learning Objectives

I can use tree diagrams to list all possible outcomes.
I can find all possible combinations.
I can draw a Venn diagram to represent the intersection and union of sets of numbers.

Introduction

The Ferris Wheel

Maggie, Sarah, and Julie are excited to go on the Ferris wheel at Valleyfair. There isn’t any line, and so the friends decide to ride the Ferris wheel multiple times in a row. It is great! The Ferris wheel stops at the top and they can see all across the entire park. Sarah spots their teacher Mrs. Martinez and gives a huge wave. The others join in.

Each seat can only hold two people so the friends take turns sitting with each other. They keep riding the Ferris wheel until everyone has had a chance to sit with everyone else. After the last ride, they get off the ride, a little dizzy, but very happy!

“Wow that was some time!” Maggie says excitedly.

“Yes, but my head is still spinning,” Julie declares.

As they walk away, Chris comes over. When he asks where they have been, they tell him that they have been riding the Ferris wheel.

“How many times did you ride it?” Chris asks.

All three of the friends look at each other. They aren’t sure. It was so exciting to keep riding that they lost count.

“I know we can figure this out mathematically,” Maggie says to the others as she starts to count on her fingers.

Do you know how many times they rode the Ferris wheel? If each friend rode with each other once, how many times did they ride in all? You can use a few different methods to figure out this outcome. In this lesson, you will learn all about finding outcomes. Pay attention so that you can figure this problem out in the end.

Guided Learning

Use Tree Diagrams to List all Possible Outcomes

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. We can also think about chances or probability through calculating outcomes.

What is an outcome?

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. We do this all the time in life and we don’t even realize that we are doing it. Anytime you are trying to organize something with many different pieces or components, you are figuring outcomes.

How can we figure out an outcome?

There are a couple of different ways to do this, and you are going to learn about them in this lesson. The first one that we are going to work with is a tree diagram.

What is a tree diagram?

A tree diagram is a visual way of listing outcomes. You look at the choices for the outcome and the variables that go with each outcome.

Alright, let’s slow down. First, let’s look at an example and that will help make tree diagrams a lot clearer.

Example A

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, we are going to create a tree diagram.

First, we list the choices of ice cream.

Vanilla Crunch

Black Raspberry

Chocolate Chip

Lemonade

Next we 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. Tree diagrams show all the possibilities. It doesn’t not matter which you start with, the ice cream flavor or the type of cone. You will end up with the same number of possibilities. The second tree diagram begins with the cone flavor. You will end up with the same possibilities.

Screen Shot 2014-06-02 at 9.24.27 PM.png

Here we have four different flavors, and two types of cones, which means we have 8 possible ice cream cone options: sugar cone with vanilla crunch, sugar cone with black raspberry, sugar cone with chocolate chip, sugar cone with lemonade, plain cone with vanilla crunch, plain cone with black raspberry, plain cone with chocolate chip, and a plain cone with lemonade ice cream.

Tree diagrams can be helpful to identify the possible outcomes and find the probability of an event occurring.

Example B

If you toss a coin two times, what is the probability of getting two heads? Use a tree diagram to find your answer.

When you flip the coin once, you have an equal chance of getting a head (H) or a tail (T). On the second flip, you also have an equal chance of getting a head or a tail. In other words, whether the first flip was heads or tails, the second flip could just as likely be heads as tails. You can represent the outcomes of these events on a tree diagram.

From the tree diagram, you can see there are four possible outcomes. You can also see that the probability of getting two heads is 1 out of 4 possible outcomes. Therefore, we can conclude that the probability of getting two heads when tossing a coin twice is $\frac{1}{4}$ , or 25%.

Example C

Irvin opens up his sock drawer to get a pair of socks to wear to school. He looks in the sock drawer and sees three pairs of socks: red socks, white socks, and brown socks. What are the possible combinations of socks Irvin could pull out if he picks one pair, returns the first pair and chooses again?

First let’s draw a tree diagram. There are 9 possible combinations of socks.

Irvin reaches in the drawer and pulls out red socks. He is wearing blue shorts, so he replaces them. He then draws out white socks. What is the probability of getting red socks, then white socks? Looking at the results of the tree diagram, we can determine there is a 1 in 9 chance or 11% chance that Irvin will pull out first a pair of red socks, then a pair of white socks.

What is the probability that Irvin will choose a pair of white socks if he tries picking socks twice? Looking at the tree diagram, we can see 5 possible outcomes out of 9 that include a pair of white socks: red/white, white/red, white/white, white/brown, and brown/white. That means Irvin has a 5/9 or 56% chance of choosing white socks if he has two chances to choose.

Find All Possible Combinations

When you have a combination, order does not matter. The ice cream cones were a good example. It did not matter what the order was of the flavors or the cones. We just wanted to know how many different possible cones could be created.

We can find all of the possible combinations when working with examples.

How do we do that?

We work on figuring out combinations by listing out all of the possible options. Then we eliminate any duplicates and the number of outcomes left is our answer. Let’s look at an example.

Example D

Seth, Keith, Derek, and Justin want to go on the bumper cars. They can only ride in pairs. How many different paired combinations are possible given these parameters?

To start, we list out all possible options beginning with Seth. Seth can ride with Keith, Derek, or Justin. Keith can ride with Seth, Derek, or Justin. Derek can ride with Seth, Justin, or Keith. Justin can ride with Seth, Derek, or Keith.

Here are the possible outcomes. A permutation is a combination where order makes a difference. If order mattered, this would be our list of outcomes. You will learn more about permutations in the future.

$&\text{SK} && \text{KS} && \text{DS} && \text{JS}\\&\text{SD} && \text{KD} && \text{DK} && \text{JK}\\&\text{SJ} && \text{KJ} && \text{DJ} && \text{JD}$

To find the possible combinations, we cross out any duplicates.

$&\text{SK} && \text{\bcancel{KS}} && \text{\bcancel{DS}} && \text{\cancel{JS}}\\&\text{SD} && \text{KD} && \text{\cancel{DK}} && \text{\cancel{JK}}\\ &\text{SJ} && \text{KJ} && \text{DJ} && \text{\cancel{JD}}$

There are six different pair combinations for these students to ride the bumper cars.

Guided Practice

Practice finding outcomes. Create a tree diagram to find the possible outcomes.

Marcela has three pairs of pants and four shirts. How many different outfits can she create with these choices?
Travis has four different pairs of striped socks (blue stripes, red stripes, green stripes, and orange stripes) and two pairs of sneakers, one red and one blue. How many different shoe/sock combinations can Travis create? What is the probability that Travis will wear the blue striped socks and blue shoes?

Take a few minutes to compare your work with a friend’s work. Are there any differences in your approaches? How about in your answers?

Solutions

1. 2. Probability of blue striped socks with

blue shoes = ⅛, or 12.5%

Screen Shot 2014-06-03 at 8.56.55 PM.png

Venn Diagrams

In probability, a Venn diagram is a graphic organizer that shows a visual representation for all possible outcomes of an experiment and the events of the experiment in ovals. Normally, in probability, the Venn diagram will be a box with overlapping ovals inside. Look at the diagram below:

The rectangle represents all of the possible outcomes of an experiment. It is called the sample space. The ovals and represent the specific outcomes of the events and that occur within the sample space. If an outcome occurs in both events and , that outcome will be represented in the portion of the ovals that overlap each other.

Example D

Event represents randomly choosing a student from Washington High School who holds a part-time job. Event represents randomly choosing a student from Washington High School who is on the honor roll. Draw a Venn diagram to represent this example.

We know that the sample space will include all students of Washington High School. Event includes the students with a part-time job, and event includes the students on the honor roll.

S = {all students in Washington High School}

A = {students holding a part-time job}

B = {students on the honor roll}

Notice that the overlapping oval for and represents the students who have a part-time job and are on the honor roll. The sample space, , outside the ovals represents students neither holding a part-time job nor on the honor roll. While we don't the how many students are included in each event, the Venn diagram does show how they could be related.

Example E

You are asked to roll a die. Event A is the event of rolling a 1, 2, or a 3. Event B is the event of rolling a 3, 4, or a 5. Draw a Venn diagram to represent this example. What would the solution set for A or B look like? What would the solution set for A and B look like?

We know the sample space (all possible outcomes) for rolling a die is {1, 2, 3, 4, 5, 6}. We also know Events A and B because they are given in the problem.

$S &= \{1, 2, 3, 4, 5, 6\}\\A &= \{1, 2, 3\}\\B &= \{3, 4, 5\}$

Listing the sample space with Events A and B help us to identify which numbers will be located in which spaces.

The only number not included in events A or B is 6. It is part of the Sample Space, but not part of an event so it should be located in the rectangle, but not in an event oval.
Events A and B have one number, 3, in common. That number needs to be displayed in the portion of the ovals that overlap since it occurs in both events.
Other numbers that occur in only Event A or Event B should be displayed in their respective oval spaces.

Back to the questions:

What would the solution set for A or B look like? You need to look at all outcomes that are in both ovals A and B and include them in the answer. Solutions for A or B are {1, 2, 3, 4, 5}. There are 5 numbers that occurred in Event A or Event B.

What would the solution set for A and B look like? You need to look only at the outcomes that are in BOTH A and B, the part of the ovals that overlap. Solutions for A and B are {3}. There is only 1 number that occurred in both Event A and Event B.

Guided Practice

1. 2 coins are tossed one after the other. Event A consists of the outcomes when tossing heads on the first toss. Event B consists of the outcomes when tossing heads on the second toss. Draw a Venn diagram to represent this example.

Check your work with a partner.

Solution

1. The first step is to identify all possible outcomes, the sample space. You can create a tree diagram for this or list them another way. We know that there will be four possible outcomes shown by S below. Results for events A and B are listed below as well.

$S &= \{HH,HT, TH, TT\}\\A &= \{HH,HT\}\\B &= \{HH,TH\}$

Real Life Example Completed

The Ferris Wheel

Here is the original problem once again. Reread it and underline any important information.

Maggie, Sarah, and Julie are excited to go on the Ferris wheel. There isn’t any line, so the friends decide to ride the Ferris wheel multiple times in a row. It is great! The Ferris wheel stops at the top and they can see all across the entire park. Sarah spots their teacher Mrs. Martinez and gives a huge wave. The others join in.

Each seat can only hold two people, so the friends take turns sitting with each other. They keep riding the Ferris wheel until everyone has had a chance to sit with everyone else. After the last ride, they get off the ride, a little dizzy, but very happy!

“Wow that was some time!” Maggie says excitedly.

“Yes, but my head is still spinning,” Julie declares.

As they walk away, Chris comes over. When he asks where they have been, they tell him that they have been riding the Ferris wheel.

“How many times did you ride it?” Chris asks.

All three of the friends look at each other. They aren’t sure. It was so exciting to keep riding that they lost count.

“I know we can figure this out mathematically,” Maggie says to the others as she starts to count on her fingers.

Thinking about tree diagrams, Venn diagrams, and combinations, how can Maggie figure this out mathematically?

We could use a tree diagram to figure this out. We could also write out all of the combinations.

In this combination we have three friends sitting two at a time: Maggie and Sarah, Maggie and Julie, Sarah and Maggie, Sarah and Julie, Julie and Maggie, Julie and Sarah.

MS MJ SM SJ JM JS

Next, we need to cross out the duplicates.

MS MJ SM SJ JM JS

They each went for three rides on the Ferris Wheel in a row!

Wow! No wonder they were dizzy!

Review

Tree Diagrams can be used to list possible outcomes.
Create a tree diagram to find the possible outcomes from events.
Construct venn diagrams from the sample space and event outcomes. Compare events to determine if there will be solutions that overlap.
Combinations include all possibilities when order does not matter.

Probability

The probability that something will happen are the chances that something will happen. It can be written as a fraction, decimal, or percent.

Outcome

The end result is called the outcome.

Sample Space

The set of all possible outcomes is called the sample space of the experiment.

Tree Diagram

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

Venn diagram

A Venn diagram is a diagram of overlapping circles that shows the relationships among members of different sets.

Combination

A combination is an arrangement of options where order does not make a difference.

Additional Resources

Creating Tree Diagrams Video

Creating a Venn Diagram Video