Lesson 3

Sample Spaces

3.1: Rolling Cubes (5 minutes)

Warm-up

The mathematical purpose of this activity is to represent sample spaces for compound events.

Launch

Arrange students in groups of 2. Tell students that standard number cubes will be referenced throughout this unit. A standard number cube is a cube with each side labeled with one of the numbers 1 through 6. Tell them that “1 on the first cube and 3 on the second cube is different from 3 on the first cube and 1 on the second cube.” If possible, roll two different (by color or size) number cubes to demonstrate. Monitor for students who use an organized list, a table, or a tree diagram to record the sample space.

Student Facing

When rolling two standard number cubes, one of the possible outcomes is 1 and 1.

1. What are the other possible outcomes?
2. How many outcomes are in the sample space?

Activity Synthesis

The purpose of this discussion is to help students think about sample space and to have students share any methods that they used to find the sample space. Ask previously identified students to share their representation with the class. If students did not use an organized list, table, or tree diagram they will encounter the three representations later in this lesson.

Here are some questions for discussion.

• “What strategy did you use to make sure you found all of the outcomes?” (I used an organized list to make sure that I did not miss any of the outcomes.)
• “What strategy makes more sense for you to use?” (I thought that the organized list was easier to use. I started with a tree diagram and I noticed a pattern that made it easy for me to create an organized list.)
• “How did you make sure you found all of the outcomes without repeating any?” (I knew that there were 36 total outcomes so I just made sure I did not have any repeats in the 36 outcomes that I wrote down.)
• “What is the probability of rolling 2 on the first die and a 2 on the second die?” ($$\frac{1}{36}$$)
• “What is the probability that the first die rolled is a 5?” ($$\frac{6}{36}$$)

3.2: Spinner Sample Space (10 minutes)

Activity

The mathematical purpose of this activity is to compare different methods that can be used to represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams.

Launch

Arrange students in groups of 2. Tell students that the activity involves comparing three different methods for finding the sample space of an experiment.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret the language of probability, and to increase awareness of language used to discuss probabilities. Display only the task statement, spinners, and Diego’s, Tyler’s, and Lin’s different representations of the probability. Ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the remainder of the questions. Listen for and amplify any questions involving probabilities and different ways to represent the outcomes.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

Each of the spinners is spun once.

• Diego makes a list of the possible outcomes: ALW, ALX, ALY, ALZ, AMW, AMX, AMY, AMZ, ANW, ANX, ANY, ANZ, BLW, BLX, BLY, BLZ, BMW, BMX, BMY, BMZ, BNW, BNX, BNY, BNZ
• Tyler makes a table for the first two spinners.

L M N
A AL AM AN
B BL BM BN

Then he uses the outcomes from the table to include the third spinner.

W X Y Z
AL ALW ALX ALY ALZ
AM AMW AMX AMY AMZ
AN ANW ANX ANY ANZ
BL BLW BLX BLY BLZ
BM BMW BMX BMY BMX
BN BNW BNX BNY BNZ
• Lin creates a tree to keep track of the outcomes.
1. How many outcomes are in the sample space for this experiment?
2. One of the outcomes from Diego’s list is BLX. Where does this show up in Tyler's method? Where is it in Lin’s method?
3. When spinning all three spinners, what is the probability that:
1. they point to the letters ANY? Explain your reasoning.
2. they point to the letters AMW, ANZ, or BNW? Explain your reasoning.
4. If a fourth spinner that has 2 equal sections labeled S and T is added, how would each of the methods need to adjust?

Anticipated Misconceptions

Some students may create a tree diagram but may not understand how to quantify the number of outcomes in the sample space. Prompt students to look at their tree diagram and to count each individual outcome. Show students that each branch at the end of the tree represents an outcome in the sample space.

Activity Synthesis

The purpose of this discussion is for students to understand how each of the three representations can be used to represent the same sample space.

Here are some questions for discussion.

• “How many cells will there be in Tyler's method for each table?” ($$2 \boldcdot 3$$ first, then $$(2 \boldcdot 3) \boldcdot 4$$)
• “How many paths are in Lin’s method?” (The first row has 2 groups of 1 thing each. Second row has 2 groups of 3 things each. Third row has 6 groups of 4 things each.)
• “Which method do you think you would use to solve this problem? Explain your reasoning.” (I would use the organized list. I just start with the sample space from the first spinner and then add to the list using the sample space from the second spinner, and so on. It is similar to a tree diagram, but I don’t actually create the tree diagram on paper.)

If time permits display each representation for all to see.

Organized list: ALW, ALX, ALY, ALZ, AMW, AMX, AMY, AMZ, ANW, ANX, ANY, ANZ, BLW, BLX, BLY, BLZ, BMW, BMX, BMY, BMZ, BNW, BNX, BNY, BNZ

Table:

W X Y Z
AL ALW ALX ALY ALZ
AM AMW AMX AMY AMZ
AN ANW ANX ANY ANZ
BL BLW BLX BLY BLZ
BM BMW BMX BMY BMX
BN BNW BNX BNY BNZ

Tree diagram:

Here are some questions for discussion.

• “What are the benefits of each representation?” (The organized list summarizes the same information that is in the table and the tree diagram and it is easy to see all of the information at once. The table organizes the information so that you can see where it comes from. The tree diagram helps you to see that you have all of the possibilities.)
• “Why wouldn’t you use one representation over another?” (I prefer to use the organized list over the tree diagram because it is hard for me to see each of the combinations in the tree diagram. However, sometimes I do use the tree diagram to help me create my organized list.)
• "How could you determine the new number of outcomes without writing out the sample space when we add the fourth spinner?" (I would double the number of outcomes because I know that there are two additional possibilities for each existing outcome.)

3.3: Sample Space Practice (15 minutes)

Activity

The mathematical purpose of this activity is for students to record the sample space for experiments involving multiple events and to explore different methods for recording sample spaces. Identify students who use an organized list, table, or tree diagram. Monitor for students who:

• do not use an organized list, table, or tree diagram
• use the same type of representation for each of the experiments
• use more than one type of representation
• use different types of representations for a single experiment

Launch

Arrange students in groups of 2. Tell students:

• “All coins and number cubes mentioned in this unit are assumed to be fair unless otherwise noted. This means that there is a an equal chance for each of the outcomes to result.”
• “After completing each question, compare your solution and your solution method to those of your partner.”

Allow students a few minutes to think of solutions on their own, before sharing with their partner.

Conversing: MLR8 Discussion Supports. Use this routine to support small-group discussion. Ask students to take turns describing the sample space for each situation and explaining their reasoning to their partner. Display the following sentence frames for all to see: “The outcomes are _____ because . . .”, “I chose this representation because . . .”, and “I noticed _____, so I thought . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their reasoning of sample spaces.
Design Principle(s): Support sense-making; Maximize meta-awareness
Action and Expression: Internalize Executive Functions. Provide students with blank graphic organizers to choose from that can be used to support thinking. For example, blank tables with 4 columns and 5 rows, and blank tree diagrams to match each scenario.
Supports accessibility for: Language; Organization

Student Facing

List all the possible outcomes for each experiment.

1. A standard number cube is rolled, then a coin is flipped.
2. Four coins are flipped.
3. The two spinners are spun.
4. A class block is chosen from 1, 2, 3, 4, or 5, then a subject is chosen from English or math.

Student Facing

Elena is answering a matching practice problem where she has to match each of 4 items in a left column (A, B, C, D) to 4 items on the right (1, 2, 3, 4) so that each item is used exactly once. For example one way to answer this problem is A4, B2, C1, D3.

1. What are all the possible ways she could answer this problem?
2. The actual solution is A1, B2, C3, D4. If Elena was equally likely to guess any answer, use the sample space to find:
1. The probability of getting exactly 0 items matched correctly.
2. The probability of getting exactly 1 item matched correctly.
3. The probability of getting exactly 2 items matched correctly.
4. The probability of getting exactly 3 items matched correctly.
5. The probability of getting exactly 4 items matched correctly.

Activity Synthesis

Select previously identified students to share in this order:

• do not use an organized list, table, or tree diagram
• use the same type of representation for each experiment
• use more than one type of representation
• use different types of representations for a single experiment

The purpose of this discussion is for students to analyze different methods for recording sample spaces and to determine probabilities from a sample space using different representations. Here are some questions for discussion.

• "Why did you choose to use the same representation each time?" (I remember using tree diagrams in middle school, so I just used them again and again.)
• "Why did you choose to use different representations?" (After using the tree diagram twice, I realized that I could make an organized list quickly without having to make the tree diagram.)
• "Why did you use more than one representation for a single experiment?" (I used a tree diagram, but then I organized my results in a table. I thought that the table would help me to see the information more easily.)
• "What are the advantages of using one representation over another?" (I thought that the tree diagram was the easiest way for me to make sure that I listed all of the possibilities. However, it is difficult to read the different possibilities in a tree diagram so using an organized list is an easier way to summarize the same information.)
• “For the experiment in question one, what is the probability that the outcome is heads? Explain your answer using the representation you used.” (The probability is $$\frac{1}{2}$$. I made a tree diagram and for every roll of the number cubes, half of the outcomes had heads and half of them had tails.)
• “For the experiment in question two, what is the probability that the outcome is three heads and one tail? Explain your answer using the representation you used.” (The probability is $$\frac{4}{16}$$. I created an organized list. After writing HHHH, I recorded all the possible outcomes that had exactly three heads. There were four of them.)
• “For the experiment in question three, what is the probability that the outcome is R1, R2, G1 or G2? Explain your answer using the representation you used.” (The probability is $$\frac{4}{20}$$. I looked in my table and each of the 4 outcomes listed only appeared one time.)

Lesson Synthesis

Lesson Synthesis

Here are some questions for discussion.

• “How do you use the sample space to calculate probabilities?” (The sample space lists all of the possible outcomes, so you can calculate the probability directly from the sample space.)
• “How does using a tree diagram, organized list, or table help you to answer questions about probability?” (It helps because if you list all of the possible outcomes then you can calculate the probability. It also helps because you can see patterns emerge while creating a tree diagram, organized list, or table. Sometimes you can find the probability using the pattern, instead of having to list all of the possibilities. For example, when I made a tree diagram for the number cubes, I found that when the first number cube was 1 that there were six possibilities for the second cube. There were the same six possibilities when the number cube was 2, 3, 4, 5, and 6.)
• “Ask your partner a probability question about the experiment with two spinners in question 4 of the last activity. What question did you ask or what question were you asked? What is the answer to the question?” (I asked my partner, “What is the probability of getting 1 and math?” and the answer is 0.1.)

Student Lesson Summary

Student Facing

Probability represents the proportion of the time an event will occur when repeating an experiment many, many times. For complex experiments, the sample space can get very large very quickly, so it is helpful to have some methods for keeping track of the outcomes in the sample space.

In some cases, it makes sense to list all the outcomes in the sample space. For example, when flipping 3 coins, the 8 outcomes in the sample space are:

HHH, HHT, HTH, THH, HTT, THT, TTH, TTT

where H represents heads and T represents tails.

With more outcomes possible, it can be difficult to make sure all the outcomes are represented and none are repeated, so other methods may be helpful.

Another option is to use tables. When a complex experiment is broken down into parts, tables can be used to find the outcomes of two parts at a time. For example, when flipping 3 coins, we determine the outcomes for flipping just 2 coins.

H T
H HH HT
T TH TT

The possible outcomes are represented by the 4 options in the middle of the table: HH, HT, TH, and TT. These outcomes can then be combined with the third coin flip in another table.

H T
HH HHH HHT
HT HTH HTT
TH THH THT
TT TTH TTT

Again, we see that the outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT.

Another way to keep track of the outcomes is to draw a tree structure. Each column represents another part of an experiment, with branches connecting each possible result from one part of the experiment to the possible results for the next part. By following the branches from left to right, each path represents an outcome for the sample space. The tree for flipping 3 coins would look like this:

The path shown with the dashed line represents the HTH outcome. By following the other paths, the other 7 outcomes can be seen.