Unit II Probability Section A: Determining Probabilities

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1 Section Planning Learning Outcomes Unit II Probability Section A: Determining Probabilities 1. analyze and construct representations of events, including tree diagrams, to determine conditional probabilities. 2. construct Venn diagrams and determine probabilities of compound events to make a decision about the risks involved in the situation. 3. analyze and construct area models to determine probabilities of events in order to make decisions about the risks involved in problem situations. Student Expectations (DM.2) Analyzing information using probability. The student analyzes and evaluates risk and return in the context of everyday situations. The student is expected to: (A) (B) determine conditional probabilities and probabilities of compound events by constructing and analyzing representations (including tree diagrams, Venn diagrams, and area models) to make decisions in problem situations; use probabilities to make and justify decisions about risks in everyday life (such as investing in the stock market, taking medication, or selecting car insurance); and Time Management This section is designed to take four instructional days (45-minute class periods). This estimate does not include any class time dedicated to extension or reflection questions. Lesson II.A.1 deals with analyzing and constructing Venn diagrams to determine probabilities of events in order to make decisions in problem situations. Conditional probability is defined during this lesson. This lesson is supported by Student Activity Sheet 1. Lesson II.A.2 deals with analyzing and constructing tree diagrams to determine probabilities of events in order to make decisions in problem situations. also investigate compound probability. Dependent and independent events are discussed during the lesson. This lesson is supported by Student Activity Sheet 2. II-3

2 Lesson II.A.3 deals with analyzing and constructing area models to determine probabilities of events in order to make decisions about the risks involved in problem situations. This lesson is supported by Student Activity Sheet 3. Prerequisite Skills Understanding how to determine probability of simple events Understanding the concept of equally likely Understanding experimental versus theoretical probability Vocabulary Academic Vocabulary: area model, complement sets, compound probability, conditional probability, dependent events, equally likely, independent events, probability, sample space, tree diagram, Venn diagram Application Vocabulary: random Materials Markers (assorted colors) Poster paper Paper bags Colored cubes Colored marbles Related Resources None Additional Background Connections should come into this course with an understanding of basic probability, including the concept that probabilities are useful for predicting what might happen. need to be able to interpret statements of probability to make decisions or answer questions. II-4

3 Instructional Strategies This section offers multiple opportunities for students to work in pairs or small groups, but students also need individual think time before beginning group work. Encourage students to look at solving problems in a variety of ways; have discussions to solidify new learning. Things to Watch for should have a firm understanding of the probability of simple events; take this opportunity to review as students begin to build their understanding of more complex probability. also need to know what and and or represent in set notation. often get lost when reading tree diagrams. If this happens, suggest to students that they list the sample space (that is, write down all possible outcomes for the event). II-5

4 Lesson II.A.1: Using Venn Diagrams Opening the Lesson Have students make a Venn diagram using whole numbers less than or equal to 20 and even whole numbers less than or equal to 20. Framing Questions How do you determine which numbers to put in the parts of the Venn diagram? Activities During the lesson, students work in small groups and then come together for wholeclass discussions. on their own before talking to group members. individually on questions that check for understanding. In whole-class discussion, ask students to summarize the main ideas from Questions 1 4 before they move on to Questions 5 8. II.A.1 Student Activity Sheet 1 Question 1: work in small groups to generate answers. Whole-class discussion. 1. Begin the lesson by discussing Venn diagrams. It is likely that students have used them before. Venn diagrams are one way to organize information given in probability situations. Ask the following about the Venn diagram before Question 1: What does the rectangle in the Venn diagram represent? (the universal set) What do the circles represent? (the various events associated in the problem) 2. Ask students to study the Venn diagram. What do the different parts of the Venn diagram tell you? Following are examples of facts that students might list: There are 29 students in the class. Six students are not taking Algebra II or Chemistry. Four students are taking Algebra II and Chemistry. Nine students are taking Chemistry. Eighteen students are taking Algebra II. Of the 18 students taking Algebra II, 4 of them are also taking Chemistry. 3. Have students list their facts. Keep a class list of all examples. As correct examples are shared, students add them to their own list. II-6

5 Questions 2 and 3: individually for a short time and then in small groups to talk about results. Whole-class discussion about students understanding of probability. Question 4: work in small groups. 1. Ask students the following questions: How is a probability written? (in fraction form, as a decimal, or as a percentage) What does it mean when you take the probability of two things? (In this situation, it means you want the probability of selecting a student who is taking Algebra II and Chemistry the intersection of the two groups.) Where on the Venn diagram do you find the students who are not taking Algebra II? (Outside the Algebra II circle. In this situation, it is the 6 students outside both groups plus the 5 Chemistry-only students.) Where on the Venn diagram do you find the students who are not taking Chemistry? (Outside the Chemistry circle. In this situation, it is the 6 students outside both groups plus the 14 Algebra II-only students.) 2. Discuss complement sets in probability. What is the complement of not taking Algebra II or not taking Chemistry? (taking Algebra II and taking Chemistry) 1. Ask the following: What does it mean to select a student who is taking Algebra II or Chemistry? (There are 29 students in Ms. Snow s homeroom, and 6 students are not taking Algebra II or Chemistry. Therefore, 29 6 = 23, so the probability of students taking Algebra II or Chemistry is 23/29. A second way to think about this situation is 14 students taking only Algebra II plus 5 students taking only Chemistry plus 4 students taking both. Therefore, the probability of students taking Algebra II or Chemistry is 23/29. Another way to approach this situation is to take the probability of students taking Algebra II (18/29) plus the probability of students taking Chemistry (9/29) to get 27/29, minus the intersection of the two (4/29) to arrive at the probability of 23/29.) II-7

6 Question 5: in pairs. Scenario Introduction: individually at first and then discuss the diagrams with each other and as a class. Questions 6-8: individually. Questions 9-10: individually. 1. Before students begin work on Question 5, define conditional probability. (This is the probability of Event B given that Event A has already occurred or is certain to occur. The symbols P[B/A] denote this process of finding the probability of a dependent event [in this case, Event B]. Event B is conditional on Event A having occurred.) 2. As students work, be sure that they are considering only the reduced number of possibilities from the class. This is the critical difference with conditional probability. 1. Say the following: Earlier we started with a Venn diagram, and you answered questions about it. Now you will take some information and construct a Venn diagram. 2. Ask the following: What are the two groups in the Venn diagram? (males and two-handed backhand players) What is the sum of the numbers in your Venn diagram? (758) Where do you find the females with a two-handed backhand? (in the part of the circle of the two-handed backhand players that does not intersect the circle of the males) 3. Have the different student groups put their Venn diagrams on poster paper. Compare and contrast these diagrams as a class. 1. After students are sure that their Venn diagrams are correct, have them work on Questions 6-8. Check their notations and form of listing the probabilities. You can continue to ask questions about other probabilities or complements. Make sure students can explain their reasoning. 1. Question 9 is similar to Question 4. Have students share the different ways they found the answer to this question. 2. Similarly, Question 10 is like Question 5, using conditional probability. Have students share their reasoning. Further Questions How does a Venn diagram help you find probability? What are some limitations of a Venn diagram? II-8

7 Lesson II.A.2: Using Tree Diagrams Opening the Lesson Say the following: You have just learned to use a Venn diagram to help answer probability questions. Some of these situations involved conditional probability. What other methods can help you visualize the sample space? Some probability situations have multistage events. Tree diagrams help show the stages of events, compound probability, and all the possible outcomes. Framing Questions What is a tree diagram? How are the stages of an event shown on a tree diagram? How do you read a tree diagram? Activities During the lesson, students work in small groups and then come together for wholeclass discussions. on their own before talking to group members. individually on questions that check for understanding. In whole-class discussion, ask students to summarize the main ideas from Questions 1 7 before they move on to the rest of the questions. II.A.2 Student Activity Sheet 2 Question 1: individually on tree diagrams and then discuss outcomes in groups and as a whole class. 1. Have students individually construct a tree diagram on their activity sheet and then discuss their diagrams in small groups. Give one group poster paper for transferring its tree diagram. Compare and contrast the poster with students individual diagrams. Have students correct their diagrams if needed. 2. Ask questions to make sure students understand the tree diagram. How many possible outcomes occur in the sample space? (24) Do you see a relationship between the number of choices in each category in the table and the number of total outcomes? (In the table, there are 2 choices of bread times 3 choices of meat times 4 choices of cheese, which produces 24 possible outcomes.) II-9

8 Questions 2 and 3: individually. Questions 4 7: in small groups and then come together for whole-class discussion. Question 8: work in small groups. 1. should be able to answer these questions individually. As you monitor their work, ask students to explain their thinking. These questions are a review of prior work. 1. Review the definition of conditional probability. Ask the following questions to make sure students understand the term: What happens when we eliminate one bread choice? In other words, we ask questions that are given white bread. (We eliminate half of the outcomes.) What was the probability of getting white bread (originally)? (1/2 or 50 50) What was the probability of either Swiss or American cheese in a sandwich? (12/24) What is now the probability of either Swiss or American cheese in a sandwich? (6/12) 2. Review the term equally likely with students. When are events equally likely? (Events are equally likely when two or more events have the same probability of occurring. When you toss a fair coin, the chance that it lands on tails or heads is equally likely. Hence, it is a fair coin.) Are the events in this problem equally likely? (yes) 3. Discuss the dependent and independent natures of an event. Are the events in this situation dependent or independent? (independent) What does it mean for an event to be dependent? (The outcome of an event affects the outcome of a second event; for example, selecting a coin from a bag and then selecting another coin from the same bag without replacing the first coin narrows the possibility of what coin will be chosen in the second event.) 1. Explain how using a tree diagram helps you analyze the probability of an event. (Tree diagrams help you find all the possible outcomes of a situation. In this situation, you are interested in all the possible groups that can go with Catrina to visit the middle schools.) Is there another way to find how many outcomes Catrina will have? (5 boys 4 girls = 20 outcomes; this calculation tells you how many outcomes but does not list all the possible outcomes.) II-10

9 Questions 9 and 10: work in small groups. Question 11 Question 12: independently. Questions 13 16: individually, but allow group discussion before having a class discussion. 1. As students work in groups, ask them to explain equally likely. 1. Ask the following: How does changing the number of girls affect the outcomes? (It affects the outcomes for the girls, but not for the boys. With Ave going, each girl has a 1/4 probability of being selected. With Ave not going, each girl has 1/3 probability of being selected. With Ave going, Nathan had a 4/20 or 1/5 probability of going. With Ave not going, Nathan has a 3/15 or 1/5 probability of going.) 1. Have students exchange scenarios, answer each other s questions, and discuss their work. Have a couple of students share with the class. 1. As students begin to evaluate the different group suggestions, ask questions to make sure they are finding the correct probabilities. Finding the probabilities and their complements helps students sort out the events. 2. can use a variety of ways to find the probabilities needed. Some might read the branches of the tree diagram and keep track of the outcomes. Others might use numeric reasoning. A student might suggest the following for Group 1: When you select from the first bag, you have a 3/3 chance of selecting red, white, or blue. Once you have selected a color from the first bag, you have a 2/3 chance of selecting the right color from the second bag. When selecting from the third bag, you have a 1/3 chance of selecting the color needed; therefore, you have 3/3 2/3 1/3 = 6/27 or 2/9 probability of selecting a red cube, white cube, and blue cube. Further Questions Compare and contrast using Venn diagrams and tree diagrams. Create a compound probability situation with dependent events. II-11

10 Lesson II.A.3: Using Area Models Opening the Lesson Say the following: We have discussed different models (Venn diagrams and tree diagrams) to help us find probability. Another model to consider is called an area model, which is helpful when finding probabilities of successive events. An area model enables you to use fractions of the area of a diagram to represent corresponding probabilities. Area models are helpful when the outcomes of the events are not equally likely. The model can show that some events occupy more area than others, which indicates that these events are not equally likely. (Tree diagrams can be used to model these situations; the branches of the tree, however, must be weighted to show the likelihood of the occurrences.) Framing Questions Has anyone in this class ever drawn an area model to analyze probability situations? What do you know about the model? Activities During the lesson, students work in small groups and then come together for wholeclass discussions. on their own before talking to group members. individually on questions that check for understanding. Not all students will be familiar with area models; it is important, therefore, that they work together during this lesson. II.A.3 Student Activity Sheet 3 Questions 1 3: Whole-class discussion. 1. Ask the following: How do you begin to construct an area model to analyze probability? (Draw a square unit. We let the square unit represent the probability of 1.) What are the events in this situation? (The first event is selecting a colored marble from a jar. The second event is selecting a colored cube from another jar.) How many possibilities are there in the first event? (four, selecting a red, blue, yellow, or another red marble) Are these possibilities equally likely? (No, there are more red possibilities.) How many possibilities are there in the second event? (three, selecting a yellow, red, or green cube from the jar) Are these possibilities equally likely? (yes) II-12

11 2. To draw an area model, designate a side of the square to represent the first event. Since there are four equally likely possibilities, divide the square into four equal rectangles and label each section. 3. To represent the second event, select the side of the square perpendicular to the first and divide this section into three equal rectangles. Label each section. 4. Represent the area of the square by filling each cell with an outcome. How many outcomes are represented? (12) 5. Ask the following questions to make sure students are reading the area model correctly before they answer Questions 1 3. What does BY represent? (a blue marble and a yellow cube) Do you see any similarities among the sections of the area model? (Some sections are identical; for example, there are two RR sections and two RG sections.) Question 4: work individually. share models with the class, followed by whole-class discussion. 1. Discuss the scenario about the fundraiser. Since the area model seems different, a sample of what the area model looks like for the simple maze is provided. Ask questions using the provided area model to make sure students understand how to find probability using one. 2. Ask the following: How did Kyra find that the probability of getting a pumpkin was 2/3? (She created an area model with equal-sized parts. The upper and lower paths were already divided into two equal-sized parts. If you divide the middle path into two equal-sized parts, the whole now is divided into six equal-sized parts. Now count the parts labeled pumpkins, which is 4 of 6 parts and is equal to 2/3. Another solution: she added each part labeled pumpkin [1/2 of 1/3] + 1/3 + [1/2 of 1/3] = 1/6 + 1/3 + 1/6 = 2/6 + 1/3 = 4/6 = 2/3.) 3. Discuss the diagram of the second corn maze (the one the church will actually construct). Then ask questions about the maze to check for understanding. Do you think it is equally likely for a customer to get a pumpkin as to not get one? (Answers will vary. should explain their reasoning.) When you enter the maze, what is the first event that happens? (You must decide which of the four paths to take.) II-13

12 Are there any constraints in the problem? (Customers can only walk forward.) 4. Some students may use the concept of equal parts to find the probability of taking home a pumpkin. Others may use numerical reasoning. Make sure students have the opportunity to see both ways of reasoning. Question 5: work individually. Questions 6 and 7: work in small groups. Questions 8 and 9: work individually. 1. Check for understanding by providing Question 5 for students to work through. 1. In Question 6, an exit is defined as a path leading to the pumpkins or to no pumpkins. 2. can use the area model created for Question 4 to answer these questions, or they can draw a new area model considering only the exit paths. 3. There are a variety of answers for these questions. Make sure you address them all. 1. These questions check for understanding. Therefore, students should work individually. The mazes may look different, but the probabilities of each event are the same for each maze. Further Questions Ask students to compare and contrast the three types of models (Venn diagram, tree diagram, area model). Do some probability situations work better with some models? II-14