# Lesson 5

Combining Events

## 5.1: Notice and Wonder: Birds and Bats (5 minutes)

### Warm-up

The purpose of this warm-up is to elicit the idea that Venn diagrams can be used to represent events of a sample space, which will be useful when students describe events as a subset of a set of a sample space using characteristics of the outcomes, and as unions, intersections, or complements of other events in a later activity. While students may notice and wonder many things about these images, understanding how "and," "or," and "not" are used when discussing sample space and probability are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that traits that belong to both bats and birds are listed where the two circles overlap, all of the traits in the circles of the Venn Diagram belong to bats or birds, and the trait listed outside of both of the circles does not belong to birds or bats.

### Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the Venn Diagram for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the Venn diagram. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification or point out contradicting information. If understanding how "and," "or," and "not" are used when discussing sample space and probability does not come up during the conversation, ask students to discuss this idea.

Tell students that when we use "or" in mathematics it is inclusive. Here are some questions for discussion.

• “How many traits belong to birds or bats?” (6)
• “How many traits belong to birds and bats?” (2)
• “How many traits do not belong to birds or bats?” (1)
• “What is an “or” question that you could ask involving something in this room? How would you answer it? Tell your partner.” (I could ask, “How many people have a pencil or pen on their desk?” I would answer it by counting the number of people with a pen or a pencil on their desk.)
• “Ask a question similar to your previous one but using “and” instead of “or.” How is the question different than the previous one?” (My new question is, “How many people have a pencil and a pen on their desk?’ It is different because this would include only the students with both a pen and a pencil.)
• “Is it possible for the answer to the second question to be greater than the answer to the first question? Explain your reasoning.” (No. It is not possible because the “or” question includes all of the possibilities and the second question would have to be equal to or less than the quantity of all possibilities.)

If time permits, ask students to think about this situation. You have three people with bowls of ice cream and one has chocolate, one has vanilla, and one has both chocolate and vanilla.” Here are some questions for discussion.

• “How many people have vanilla ice cream?” (2)
• “How many people have chocolate ice cream?” (2)
• “How many people have chocolate and vanilla ice cream?” (1)
• “Is the correct answer to, ‘How many people have chocolate or vanilla ice cream’ 2 or 3? Explain your reasoning.” (It is 3 because everyone has either chocolate or vanilla?)
• “How is this related to the birds and bats questions?” (It is related because when you ask, ‘Which of these traits belong to birds or bats?’ it is everything but the 6 legs trait.)

## 5.2: Eventful Islands (10 minutes)

### Activity

The mathematical purpose of this activity is to describe events as a subset of a set of sample space using characteristics of the outcomes, and to determine the probabilities of some events using the subsets of the sample space.

### Launch

Arrange students in groups of two. Tell students, “The small dots indicate that the name listed in the diagram is a country.” Give students quiet work time and then time to share their work with a partner.

Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, provide students with multiple copies of the diagram and allow them to highlight relevant information for each of the questions.
Supports accessibility for: Visual-spatial processing

### Student Facing

The small dots next to the names indicate that the name listed in the diagram is a country.

1. Based on the categories in the Venn diagram, describe Northern Ireland in a way that will not include any other countries.
2. Based on the categories in the Venn diagram, describe the Republic of Ireland in a way that will not include any other countries.
3. How many countries displayed are not part of The British Isles?
4. How many countries displayed are part of the United Kingdom?
5. How many countries displayed are part of the Isle of Ireland?
6. How many places displayed are part of the United Kingdom and the Isle of Ireland?
7. How many places displayed are part of the United Kingdom or the Isle of Ireland?
8. If one of the crown dependencies (there are 3) is chosen at random, what is the probability that it is part of The British Isles?
9. Northern Ireland, England, Scotland and Wales are all part of the United Kingdom. If one of them is selected at random, what is the probability that it is also considered part of Great Britain?
10. Given that the Republic of Ireland, Northern Ireland, England, Scotland, Wales, and the Isle of Man are all part of The British Isles, what is the probability that one of them selected at random is part of the Isle of Ireland?

### Anticipated Misconceptions

Some students may struggle with the difference between "and" and "or" when used in a situation. Ask students in the class who play a spring sport to stand up and remain standing. Then ask students who play an instrument to stand. Tell students that "the people standing represent the people who play a spring sport or who play an instrument." Ask students who play a spring sport and who play an instrument to raise their hand. Tell students that "the people with their hands raised represent the people who play a spring sport and who play and instrument." Emphasize that "and" statements require all parts of the statement to be true and "or" statements require at least one one part of the statement to be true.

### Activity Synthesis

The goal of this discussion is to help students make the connections between subsets of sample spaces and probability, and to give them additional practice using “and,” “or,” and “not” in mathematics.

Here are some questions for discussion.

• “True or false: all the countries on the diagram are crown dependencies or in the British Isles” (True)
• “How many countries are part of the United Kingdom or are crown dependencies?” (7)
• “How many countries are part of the United Kingdom and are crown dependencies?” (0)
• “If a country is chosen at random, what is the probability that it is part of the Isle of Ireland?” ($$\frac{2}{8}$$)
Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to communicate about connections between subsets of sample spaces and probability. After students share a response, invite them to repeat their reasoning using mathematical language relevant to the lesson such as “and,” “or,” “not”, “subset”, and “sample space.” For example, say, ”Can you say that again, using the word ‘subset’?” Consider inviting the remaining students to repeat these phrases to provide additional opportunities for all students to produce this language.
Design Principle(s): Support sense-making

## 5.3: Info Gap: College and Career Planning (15 minutes)

### Activity

This info gap activity gives students an opportunity to determine information about subsets of a sample space. The subsets are then used to find a probability of an event. To complete the tables, students must ask for information from a partner using mathematical language.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

### Launch

Tell students they will continue to work with sample spaces. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles.

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to solve problems involving subsets of a sample space. Display questions or question starters for students who need a starting point such as: “Can you tell me . . . (specific piece of information)”, and “Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization

### Student Facing

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the data card:

2. Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
3. Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
4. Read the problem card, and solve the problem independently.
5. Share the data card, and discuss your reasoning.

If your teacher gives you the problem card:

3. Explain to your partner how you are using the information to solve the problem.
4. When you have enough information, share the problem card with your partner, and solve the problem independently.

### Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Here are some questions for discussion:

• “What was challenging about solving the problem?” (The challenging part was having to figure out how to use the information from my partner to fill in the table because the information only gave me clues to find the numbers rather than the numbers themselves.)

• "What percentage of students in your table are in 11th grade?" (37.2% for probroblem card 1 and 47.5% for problem card 2.)

Highlight for students how a table can be used to represent sample space and to calculate probabilities and percentages.

## 5.4: Number Cube Descriptions (15 minutes)

### Optional activity

The mathematical purpose of this activity is to compare probabilities calculated through experimentation to probabilities calculated by using the sample space. The activity is optional to offer students additional practice with creating sample spaces, calculating probability, and comparing hands-on experiments to theoretical probabilities.

### Launch

Arrange students in groups of 2. Consider demonstrating the random number generator applet for students. Each box randomly generates a number 1 through 6, simulating rolling two standard number cubes. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with whole-class discussion.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support small group discussion. After students have completed a large portion of the task, provide students with sentence frames to support their conversation comparing their answers and determining if they are both correct. For example, "I agree because . . .”, "I disagree because . . .”, “Why do you think . . . ?” This will help students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent.
Design Principle(s): Support sense-making

### Student Facing

1. Roll the double number cube simulator twenty times. Record your results in the table.

first cube second cube

first cube second cube

2. List all the possible outcomes in each event. The first one is done for you.

1. The first cube is a 6.  6,1;  6,2;  6,3;  6,4;  6,5;  6,6

2. The cubes have a 4 and a 6.

3. The cubes are doubles. (Doubles means that the number on the two number cubes is the same.)

4. The cubes are doubles and the first cube is a 6.

5. The cubes are doubles or the first cube is a 6.

6. The first cube is not a 6.

7. The cubes are doubles and the first cube is not a 6.

8. The cubes are not doubles.

3. Use the information in the table to answer the questions.

1. What percentage of the rolls have a 6 on the first cube?

2. What percentage of the rolls have a 4 and a 6?

3. What percentage of the rolls are doubles?

4. What percentage of the rolls are doubles and have a 6 on the first cube?

5. What percentage of the rolls are doubles or have a 6 on the first cube?

6. What percentage of the rolls do not have a 6 on the first cube?

7. What percentage of the rolls are doubles and do not have a 6 on the first cube?

8. What percentage of the rolls are not doubles?

4. The sample space has 36 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.

5. Why is the actual probability different from the percentage of rolls you made for each event?

### Student Facing

#### Are you ready for more?

1. Roll the triple number cube simulator forty times. Record your results in the table.

 first cube second cube third cube
first cube second cube third cube

first cube second cube third cube

first cube second cube third cube

2. List all the possible outcomes in each event. The first one is done for you.

1. The first cube is a 6.  6,6,1; 6,6,2; 6,6,3; 6,6,4;  6,6,5; 6,6,6; 6,5,1; 6,5,2; 6,5,3;  6,5,4; 6,5,5; 6,4,6; 6,6,1; 6,4,2; 6,4,3;  6,4,4; 6,4,5; 6,4,6; 6,3,1; 6,3,2; 6,3,3; 6,3,4;  6,3,5; 6,3,6; 6,2,1; 6,2,2; 6,2,3; 6,2,4; 6,2,5; 6,2,6; 6,1,1;  6,1,2; 6,1,3; 6,1,4; 6,1,5; 6,1,6.

2. The cubes have a 4, a 5, and a 6.

3. The cubes are triples. (Triples means that the number on the three number cubes is the same.)

3. Use the information in the table to answer the questions.

1. What percentage of the rolls have a 6 on the first cube?

2. What percentage of the rolls have a 4, a 5, and a 6?

3. What percentage of the rolls are triples?

4. The sample space has 216 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.

5. If you rolled 500 times, do you think that the difference between the actual probability and the percentage of rolls you made for each event would increase or decrease? Explain your reasoning.

6. Describe a method for recording the 216 outcomes in the sample space. Do not actually record all 216 outcomes.

### Launch

Arrange students in groups of 2. Distribute 2 standard number cubes to each pair of students. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.

Speaking, Representing: MLR8 Discussion Supports. Use this routine to support small group discussion. After students have completed a large portion of the task, provide students with sentence frames to support their conversation comparing their answers and determining if they are both correct. For example, "I agree because . . .”, "I disagree because . . .”, “Why do you think . . . ?” This will help students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent.
Design Principle(s): Support sense-making

### Student Facing

1. Roll two standard number cubes twenty times. Record your results in the table.

 first cube second cube
 first cube second cube
2. List all the possible outcomes in each event. The first one is done for you.

1. The first cube is a 6.  6,1;  6,2;  6,3;  6,4;  6,5;  6,6

2. The cubes have a 4 and a 6.

3. The cubes are doubles. (Doubles means that the number on the two number cubes is the same.)

4. The cubes are doubles and the first cube is a 6.

5. The cubes are doubles or the first cube is a 6.

6. The first cube is not a 6.

7. The cubes are doubles and the first cube is not a 6.

8. The cubes are not doubles.

3. Use the information in the table to answer the questions.

1. What percentage of the rolls have a 6 on the first cube?

2. What percentage of the rolls have a 4 and a 6?

3. What percentage of the rolls are doubles?

4. What percentage of the rolls are doubles and have a 6 on the first cube?

5. What percentage of the rolls are doubles or have a 6 on the first cube?

6. What percentage of the rolls do not have a 6 on the first cube?

7. What percentage of the rolls are doubles and do not have a 6 on the first cube?

8. What percentage of the rolls are not doubles?

4. The sample space has 36 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.

5. Why is the actual probability different from the percentage of rolls you made for each event?

### Student Facing

#### Are you ready for more?

1. Roll three standard number cubes forty times. Record your results in the table.

 first cube second cube third cube
 first cube second cube third cube
 first cube second cube third cube
 first cube second cube third cube
2. List all the possible outcomes in each event. The first one is done for you.

1. The first cube is a 6.  661, 662, 663, 664, 665, 666, 651, 652, 653, 654, 655, 656, 641, 642, 643, 644, 645, 646, 631, 632, 633, 634, 635, 636, 621, 622, 623, 624, 625, 626, 611, 612, 613, 614, 615, 616

2. The cubes have a 4, a 5, and a 6.

3. The cubes are triples. (Triples means that the number on the three number cubes is the same.)

3. Use the information in the table to answer the questions.

1. What percentage of the rolls have a 6 on the first cube?

2. What percentage of the rolls have a 4, a 5, and a 6?

3. What percentage of the rolls are triples?

4. The sample space has 216 outcomes. Use this and the number of outcomes in each event to find the actual probability for each event in the previous problem. Compare your answers.

5. If you rolled 500 times, do you think that the difference between the actual probability and the percentage of rolls you made for each event would increase or decrease? Explain your reasoning.

6. Describe a method for recording the 216 outcomes in the sample space. Do not actually record all 216 outcomes.

### Anticipated Misconceptions

Some students may not know how to represent probability as a percentage. Prompt students to multiply the value they obtained to represent probability (between 0 and 1 inclusive) by 100 to get the percentage. Emphasize that a percentage is a quantity described by a rate per 100.

### Activity Synthesis

The goal of this discussion is to make sure students understand that probabilities calculated using sample space and probabilities calculated by experimentation are related, but not necessarily equivalent. A secondary goal of the discussion is to informally assess student understanding of how “and,” “or,” and “not” are used when finding probabilities.

Here are some questions for discussion.

• “Why is the sample space 36 outcomes for rolling two number cubes?” (It is 36 outcomes because there are 6 choices for the first cube and 6 choice for the second cube, 6 times 6 is 36. You can see this in a tree diagram from a previous lesson.)
• “According to the sample space there are 36 different outcomes for rolling two number cubes. You only rolled the cubes 20 times. Raise your hand if you rolled the same outcome more than once when you rolled 20 times. Why is this possible?” (It is possible to repeat the same roll more than once because rolling the dice is random. Just because you rolled double sixes on one turn does not reduce your chance of rolling double sixes on the next turn.)
• “How are the events “rolling doubles” and “rolling doubles and rolling a 6 on the first cube” different? Explain your reasoning.“ (The difference between the two questions is that the first question included all of the double rolls, but the second question only counts the rolls when both cubes are six.)
• “What is the difference between the probability of “rolling doubles” and the probability of “rolling doubles and rolling a 6 on the first cube”? Explain your reason” (It is 0.15. The probability of “rolling doubles” was 0.25 and the probability of “rolling doubles and rolling a 6 on the first cube is 0.10, 0.25 minus 0.10 equals 0.15.)
• “For your 20 rolls, what percentage of them have the first die equal to 1 or 2?” (Sample response: 0.4 or 40%)
• “Using the sample space, what is the probability of rolling a 1 or a 2 on the first die?” ($$\frac{12}{36}$$ or 33.3%)

## Lesson Synthesis

### Lesson Synthesis

Here are some questions for discussion.

• “How do Venn diagrams help you to understand probability?” (Venn diagrams show relationships within the data. Sometimes a characteristic can belong to more than one category. When finding probability, it is important to understand what subset of the sample space you are trying to quantify.)
• “What is an example of a probability question that you could ask using “and,” “or,” or “not”?” (You could ask “How many people in this class play a sport and are in chorus?”)
• “Explain how the terms “and,” “or,” and “not” that we use for probability are related to Venn diagrams.” (In a Venn diagram with two overlapping circles, the region that overlaps represents the “and.” Anything outside of the circles represents the “not.” All of the information in the two circles including the region that overlaps represents the “or.”)
• “What do you find interesting or challenging about finding probabilities?” (I sometimes find it challenging to figure out exactly what the question is asking for. I find it interesting that probability is really just about representing data using relative frequencies.)

## Student Lesson Summary

### Student Facing

In many cases, it is useful to talk about the important outcomes in a sample space by naming characteristics the outcomes share or characteristics that are not in the outcomes.

For example, consider a group of 12 first-year students in college and their choices of science courses.

The circle on the left represents the students taking a chemistry class and the circle on the right represents the students taking a biology class. The region where the circles overlap represents the students taking both a chemistry class and a biology class. The students who are not included in either circle are not taking chemistry or biology.

We can describe some of the groups of students based on the characteristics they share or lack. For example:

• students A, B, C, and D are taking a chemistry course
• students A, K, and L are not taking a biology course
• students B, C, and D are taking a chemistry course and taking a biology course
• students E, F, G, H, I, and J are taking a biology course and are not taking a chemistry course
• students A, B, C, D, E, F, G, H, I, and J are taking a chemistry course or are taking a biology course

While listing the individual students in this case is not too difficult, when there are many outcomes in a sample space (which is often the case), it is often easier to describe the interesting ones using some characteristics they share.