12.4: Finding Outcomes
Introduction
The Talent Show Outfit
Alicia is going to sing for the Talent Show. She is very excited and has selected a wonderful song to sing. She has been practicing with her singing teacher for weeks and is feeling very confident about her ability to do a wonderful job.
Her performance outfit is another matter. Alicia has selected a few different skirts and a few different shirts and shoes to wear. Here are her options for shirts
Striped shirt
Solid shirt
Here are her options for skirts.
Blue skirt
Red skirt
Brown skirt
Here are her options for shoes
Dance shoes
Black dress shoes
How many different outfits can Alicia create given these options?
This is best done using a tree diagram. Alicia needs to organize her clothing options using a tree diagram. This lesson will show you all about tree diagrams. When finished, you will know how many possible outfits Alicia can create.
What You Will Learn
In this lesson you will learn how to correctly apply the following skills.
- Use tree diagrams to list all possible outcomes of a series of events involving two or more choices or results.
- Recognize all possible outcomes of an experiment as the sample space.
- Find probability of specified outcomes using tree diagrams.
Teaching Time
I. Use Tree Diagrams to List all Possible Outcomes of a Series of Events Involving Two or More Choices or Results
Nadia’s soccer team has 2 games to play this weekend. How many outcomes are there for Nadia’s team?
A good way to find the total number of outcomes for events is to make a tree diagram. A tree diagram is a branching diagram that shows all possible outcomes for an event.
To make a tree diagram, split the different events into either-or choices. You can list the choices in any order. Here is a tree diagram for game 1 and game 2.
As you can see, there are four different outcomes for the two games:
\begin{align*}\text{win-win} \quad \quad \text{win-lose}\!\\ \text{lose-win} \quad \quad \text{lose-lose}\end{align*}
What happens when you increase the number of games to three? Just add another section to your tree diagram.
In all, there are 8 total outcomes.
A tree diagram is a great way to visually see all of the possible options. It can also help you to organize your ideas so that you don’t miss any possibilities.
Example
To remodel her kitchen, Gretchen has the following choices: Floor: tile or wood; Counter: Granite or formica; Sink: white, steel, stone. How many different choices can Gretchen make?
First, let’s create a tree diagram that shows all of the possible options.
Step 1: List the choices.
Choice 1 Floor: tile, wood Choice 2 Counter: granite, formica Choice 3 Sink: white, steel, stone
Step 2: Start the tree diagram by listing any of the choices for Choice 1. Then have Choice 1 branch off to Choice 2. Make sure Choice 2 repeats for each branch of Choice 1. Can you identify the missing labels in the tree below?
Step 3: Fill in the third choice. We have left some of the spaces for you to fill in.
Step 4: Fill in the outcomes. Again some of the spaces are left for you to fill in.
You can see that there are 12 possible outcomes for the kitchen design.
II. Recognize all Possible Outcomes of an Experiment as the Sample Space
When you conduct an experiment, there may be few or many possible outcomes. For example, if you are doing an experiment with a coin, there are two possible outcomes because there are two sides of the coin. You can either have heads or tails. If you have an experiment with a number cube, there are six possible outcomes, because there are six sides of the number cube and the sides are numbered one to six. We can think of all of these possible outcomes as the sample space.
A sample space is the set of all possible outcomes for a probability experiment or activity. For example, on the spinner there are 5 different colors on which the arrow can land. The sample space, \begin{align*}S\end{align*}, for one spin of the spinner is then:
\begin{align*}S = \text{red, yellow, pink, green, blue}\end{align*}
These are the only outcomes that result from a single spin of the spinner.
Changing the spinner changes the sample space. This second spinner still has 5 equal-sized sections. But its sample space now has only 3 outcomes:
\begin{align*}S = \text{red, yellow, blue}\end{align*}
Let’s look at an example having to do with sample spaces.
Example
A small jar contains 1 white, 1 black, and 1 red marble. If one marble is randomly chosen, how many possible outcomes are there in the sample space?
Since only a single marble is being chosen, the total number of possible outcomes, the sample space, matches the marble colors.
\begin{align*}S = \text{white, black, red}\end{align*}
Sometimes, the sample space can change if an experiment is performed more than once. If a marble is selected from a jar and then replaced and if the experiment is conducted again, then the sample space can change. The number of outcomes is altered. When this happens, we can use tree diagrams to help us figure out the number of outcomes in the sample space.
Example
A jar contains 1 white and 1 black marble. If one marble is randomly chosen, returned to the jar, then a second marble is chosen, how many possible outcomes are there?
This is an example where a tree diagram is very useful. Consider the marbles one at a time. After the first marble is chosen, it is returned to the jar so now there are again two choices for the second marble. Use a tree diagram to list the outcomes.
From the tree diagram, you can see that the sample space is:
\begin{align*}S = \text{white-white, white-black, black-white, black-black}\end{align*}
12G. Lesson Exercises
What is the sample space in each example?
- A spinner with red, blue, yellow and green.
- A number cube numbered 1 – 6.
- A bag with a blue and a red marble. One marble is drawn and then replaced.
Take a few minutes to check your work with a friend.
III. Find Probabilities of Specified Outcomes Using Tree Diagrams
In the last section, you started to see how tree diagrams could be very helpful when looking for a sample space. Tree diagrams can also be helpful when finding probability.
Finding the probability of an event is a matter of finding the ratio of favorable outcomes to total outcomes. For example, the sample space for a single coin flip has two outcomes: heads and tails. So the probability of getting heads on any single coin flip is:
\begin{align*}P (\text{heads}) = \frac{favorable \ outcomes}{total \ outcomes} =\frac{1}{2}\end{align*}
You can see that the sample space is represented by a number in the total outcomes. For example, if you had a spinner with four colors, the colors by name would be the sample space and the number four would be the total possible outcomes.
What about if we flipped a coin more than one time?
To find the probability of a single outcome for more than one coin flip, use a tree diagram to find all possible outcomes in the sample space.
Then count the number of favorable outcomes within that sample space to find the probability.
For example, to find the probability of tossing a single coin twice and getting heads both times, make a tree diagram to find all possible outcomes.
The diagram shows there are 8 total outcomes and they are paired with first toss option and second toss option.
Then pick out the favorable outcome–in this case, the outcome “heads-heads” is shown in red. You could have selected any of the favorable outcomes for the probability to be accurate.
Now write the ratio of favorable outcomes to total outcomes in the sample space.
\begin{align*}P (\text{heads-heads}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{4}\end{align*}
You can see that since 1 of 4 outcomes is a favorable outcome, the probability of the coin landing on heads 2 times in a row is \begin{align*}\frac{1}{4}\end{align*}.
Let’s look at another example.
Example
What is the probability of flipping a coin two times and getting two matching results–that is, either two heads or two tails?
First, let’s create a tree diagram to see our options.
Once again, just pick out the favorable outcomes on the same tree diagram. They are shown in red.
You can see that 2 of 4 total outcomes match.
\begin{align*}P (2 \ \text{heads or 2 tails}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{2}{4}=\frac{1}{2}\end{align*}
You can see that the probability of flipping two heads or two tails is 1:2.
Real–Life Example Completed
The Talent Show Outfit
Here is the original problem once again. Reread it and then look at the tree diagram created.
Alicia is going to sing for the Talent Show. She is very excited and has selected a wonderful song to sing. She has been practicing with her singing teacher for weeks and is feeling very confident about her ability to do a wonderful job.
Her performance outfit is another matter. Alicia has selected a few different skirts and a few different shirts and shoes to wear. Here are her options for shirts
Striped shirt
Solid shirt
Here are her options for skirts.
Blue skirt
Red skirt
Brown skirt
Here are her options for shoes
Dance shoes
Black dress shoes
How many different outfits can Alicia create given these options?
This is best done using a tree diagram. Alicia needs to organize her clothing options using a tree diagram. To do this, we can take each option and create a diagram to show all of the options.
Based on this tree diagram, you can see that Alicia has twelve possible outfits to choose from.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Tree Diagram
- a visual way of showing all of the possible outcomes of an experiment. Called a tree diagram because each option is drawn as a branch of a tree
- Sample Space
- The possible outcomes of an experiment
- Favorable Outcome
- the outcome that you are looking for in an experiment
- Total Outcome
- the number of options in the sample space
Time to Practice
Directions: Use Tree Diagrams for each of the following problems.
1. The Triplex Theater has 3 different movies tonight: Bucket of Fun, Bozo the Great, and Pickle Man. Each movie has an early and late show. How many different movie choices are there?
2. Raccoon Stadium offers the following seating plans for football games:
- lower deck, middle loge, or upper bleachers
- center, side, end-zone
How many different kinds of seats can you buy?
3. Cell-Gel cell phone company offers the following choices:
- Free internet plan or Pay internet plan
- 1200, 2000, or 3000 minutes
- Premium or standard phone
How many different kinds of plans can you get?
4. Jen’s soccer team is playing 4 games next week. How many different outcomes are there for the four games?
5. The e-Box laptop computer offers the following options.
- Screen: small, medium, or large
- Memory: standard 1 GB, extra 2 GB
- Colors: pearl, blue, black
List the number of different activity choices a camper can make. Use a tree diagram to list them all. Double click to check your answers.
Directions: Answer the following questions about sample spaces.
6. What is the sample space for a single toss of a number cube?
7. What is the sample space for a single flip of a coin?
8. A coin is flipped two times. List all possible outcomes for the two flips.
9. A coin is flipped three times in a row. List all possible outcomes for the three flips.
10. A bag contains 3 ping pong balls: 1 red, 1 blue, and 1 green. What is the sample space for drawing a single ball from the bag?
11. A bag contains 3 ping pong balls: 1 red, 1 blue, and 1 green. What is the sample space for drawing a single ball, returning the ball to the bag, then drawing a second ball?
12. What is the sample space for a single spin of the spinner with red, blue, yellow and green sections?
13. What is the sample space for 2 spins of the first spinner?
14. A box contains 3 socks: 1 black, 1 white, and 1 brown. What is the sample space for drawing a single sock, NOT returning the sock to the box, then drawing a second sock?
15. A box contains 3 socks: 1 black, 1 white, and 1 brown. What is the sample space for drawing all 3 socks from the box, one at a time, without returning any of the socks to the box?
16. A box contains 3 black socks. What is the sample space for drawing all 3 socks from the box, one at a time, without returning any of the socks to the box?
17. A box contains 2 black socks and 1 white sock. What is the sample space for drawing all 3 socks from the box, one at a time, without returning any of the socks to the box?
Directions: Answer each question. Use tree diagrams when necessary.
18. What is the probability that the arrow of the spinner will land on red on a single spin?
19. If the spinner is spun two times in a row, what is the probability that the arrow will land on red both times?
20. If the spinner is spun two times in a row, what is the probability that the spinner will land on the same color twice?
21. If the spinner is spun two times in a row, what is the probability that the arrow will land on red at least one time?
22. If the spinner is spun two times in a row, what is the probability that the spinner will land on a different color both times?
23. If the spinner is spun two times in a row, what is the probability that the arrow will land on blue or green at least one time?
24. Two cards, the Ace and King of hearts, are taken from a deck, shuffled, and placed face down. What is the probability that a single card chosen at random will be an Ace?
25. If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards will be Kings?
26. If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards will match?
27. If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards NOT match?