19.1 Probability and Set Theory

Transcription

1 Locker LESSON 19.1 Probability and Set Theory ommon ore Math Standards The student is expected to: S-P.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). Mathematical Practices MP.6 Precision Language Objective Explain to a partner how to find the probability of rolling a certain number on a number cube and how to find its complement. Name lass Date 19.1 Probability and Set Theory Essential Question: How are sets and their relationships used to calculate probabilities? Explore Working with Sets set is a collection of distinct objects. Each object in a set is called an element of the set. set is often denoted by writing the elements in braces. The set with no elements is the empty set, denoted by or { }. The set of all elements under consideration is the universal set, denoted by U. Identifying the number of elements in a set is important for calculating probabilities. Use set notation to identify each set described in the table and identify the number of elements in each set. Set Set Notation Number of Elements in the Set Resource Locker ENGGE Set is the set of prime numbers less than 10. =, 3,, n () = Essential Question: How are sets and their relationships used to calculate probabilities? To calculate the probability of an event, you need to know the number of items in the set of outcomes for that event, as well as the number of items in the set of all possible outcomes. The theoretical probability of the event is the ratio of the two numbers. Set B is the set of even natural numbers less than 10. Set is the set of natural numbers less than 10 that are multiples of. The universal set is all natural numbers less than 10. B =,, 6, 8 =, 8 1,, 3,,, 6,, 8, 9 n (B) = n () = U = n ( U ) = 9 PREVIEW: LESSON PERFORMNE TSK View the Engage section online. Discuss the photograph. sk students to describe math problems that could be illustrated by the two dogs. Then preview the Lesson Performance Task. Module Lesson 1 Name lass Date 19.1 Probability and Set Theory Essential Question: How are sets and their relationships used to calculate probabilities? S-P.1 For the full text of this standard, see the table starting on page of Volume 1. Explore Working with Sets set is a collection of distinct objects. Each object in a set is called an element of the set. set is often denoted by writing the elements in braces. The set with no elements is the empty set, denoted by or { }. The set of all elements under consideration is the universal set, denoted by U. Identifying the number of elements in a set is important for calculating probabilities. Use set notation to identify each set described in the table and identify the number of elements in each set. Number of Set Set Notation Elements in the Set Set is the set of prime numbers less than 10. Set B is the set of even natural numbers less than 10. Set is the set of natural numbers less than 10 that are multiples of. The universal set is all natural numbers less than 10. =, 3,, B = 6 8,,, =, 8 1,, 3,,, 6,, 8, 9 n () = n (B) = n () = n U U = ( ) = 9 Resource HRDOVER PGES Turn to these pages to find this lesson in the hardcover student edition. Module Lesson 1 99 Lesson 19.1

2 The following table identifies terms used to describe relationships among sets. Use sets, B,, and U from the previous table. You will supply the missing Venn diagrams in the Example column, including the referenced elements of the sets, as you complete steps B I following. B D Term Notation Venn Diagram Example Set is a subset of B U 1,3,,,9 set B if every element B of is also an element,8,6 of B. The intersection of sets and B is the set of all elements that are in both and B. The union of sets and B is the set of all elements that are in or B. The complement of set is the set of all elements in the universal set U that are not in. B B or B is the double-shaded region. B is the entire shaded region. B U B U U is the shaded region. 3,, 1,9,6,8 1,9 3,,,6,8 1,,6,8,9,3,, Since is a subset of B, every element of set, which consists of the numbers and 8, is located not only in oval, but also within oval B. Set B includes the elements of as well as the additional elements and 6, which are located in oval B outside of oval. The universal set includes the elements of sets B and as well as the additional elements 1, 3,,, and 9, which are located in region U outside of ovals B and. In the first row of the table, draw the corresponding Venn diagram that includes the elements of B,, and U. See first row of table. To determine the intersection of and B, first define the elements of set and set B separately, then identify all the elements found in both sets and B. =, 3,, B =,, 6, B = 8 Module Lesson 1 PROFESSIONL DEVELOPMENT Math Background German mathematician, Georg antor ( ), is considered to be the father of set theory. antor discovered that the rational numbers are countable but the real numbers are uncountable. Two French mathematicians, Blaise Pascal ( ) and Pierre de Fermat ( ), are considered to be the founders of probability theory. The roots of probability theory lie in the letters they exchanged analyzing games of chance. EXPLORE Working with Sets INTEGRTE TEHNOLOGY Students have the option of doing the Explore activity either in the book or online. INTEGRTE MTHEMTIL PRTIES Focus on Modeling MP. Discuss how the Venn diagrams provide pictures of set relationships to help students understand the terminology. Encourage students to practice drawing Venn diagrams to use when investigating set theory. For example, discuss how a Venn diagram can make it easier to identify the complement of an intersection. VOID OMMON ERRORS Students may assume that a set that contains only 0 is the same as the empty set. ontrast the empty set, { }, with the set that contains only 0, {0}. QUESTIONING STRTEGIES What word corresponds to the intersection of two sets? Is it union? Explain. nd means the elements are in both sets, which corresponds to the intersection. Or means the elements can be in either set, which corresponds to the union. How is an intersection different from a subset? The intersection consists of the elements two sets have in common, while all of the elements of a subset lie within the set of which it is a subset. How do you know when sets overlap? Sets will overlap when they have some elements in common. Probability and Set Theory 90

3 EXPLIN 1 alculating Theoretical Probabilities INTEGRTE MTHEMTIL PRTIES Focus on Math onnections MP.1 Discuss the set notation used to define theoretical probability. onnect the notation to a word description of the probability ratio, such as, the ratio of favorable outcomes in sample space to total number of outcomes in sample space. E F G H In the second row of the table, draw the Venn diagram for B that includes the elements of, B, and U and the double-shaded intersection region. See second row of table. To determine the union of sets and B, identify all the elements found in either set or set B by combining all the elements of the two sets into the union set. B =, 3,,, 6,, 8 In the third row of the table, draw the Venn diagram for B that includes the elements of, B, and U and the shaded union region. See third row of table. To determine the complement of set, first identify the elements of set and universal set U separately, then identify all the elements in the universal set that are not in set. =, 3,, U = 1,, 3,,, 6,, 8, 9 VOID OMMON ERRORS Students may have difficulty identifying an event based on a union or intersection. Suggest that students draw Venn diagrams to model the experiment. They can begin by defining each set and then create the Venn diagram to show where the sets overlap. = 1,, 6, 8, 9 I In the fourth row of the table, draw the Venn diagram for that includes the elements of and U and the shaded region that represents the complement of. See fourth row of table. Reflect 1. Draw onclusions Do sets always have an intersection that is not the empty set? Provide an example to support your conclusion. No. Using the example sets above, = because they do not have any elements in common. Explain 1 alculating Theoretical Probabilities probability experiment is an activity involving chance. Each repetition of the experiment is called a trial and each possible result of the experiment is termed an outcome. set of outcomes is known as an event, and the set of all possible outcomes is called the sample space. Probability measures how likely an event is to occur. n event that is impossible has a probability of 0, while an event that is certain has a probability of 1. ll other events have a probability between 0 and 1. When all the outcomes of a probability experiment are equally likely, the theoretical probability of an event in the sample space S is given by P () = number of outcomes in the event number of outcomes in the sample space = _ n () n (S). Module Lesson 1 OLLBORTIVE LERNING Small Group ctivity sk each group to draw a spinner with 6 or 8 equal parts. Then ask them to use letters, colors, or numbers to distinguish each section of the spinner. Have them define two events based on the spinners. For example, if letters are used, the set of vowels and the set of letters in the word math. sk students to find the probability of each event, their complements, their union, and their intersection. Have students share their work. Review which events have a probability of 1, which have a probability of 0, and why. 91 Lesson 19.1

4 Example 1 alculate P (), P ( B), P ( B), and P ( ) for each situation. You roll a number cube. Event is rolling a prime number. Event B is rolling an even number. S = 1,, 3,,, 6, so n (S) = 6. =, 3,, so n () = 3. n () So, P () = _ n (S) = 3_ 6 = 1_. B =, 3,,, 6, so n ( B) =. So, P ( B) = _ n ( B) n (S) B =, so n ( B) = 1. So, P ( B) = _ n ( B) n (S) n ( ) = 1,, 6 Your grocery basket contains one bag of each of the following items: oranges, green apples, green grapes, green broccoli, white cauliflower, orange carrots, and green spinach. You are getting ready to transfer your items from your cart to the conveyer belt for check-out. Event is picking a bag containing a vegetable first. Event B is picking a bag containing a green food first. ll bags have an equal chance of being picked first. Order of objects in sets may vary. S = orange, apple, grape, broccoli, cauliflower, carrot, spinach, so n (S) =. = broccoli, cauliflower, carrot, spinach, so n () =. So P () = n ( ) _ n ( S ) B = _. B = broccoli, cauliflower, carrot, spinach, apple, grape, so n ( B) = 6. n ( B ) 6 P ( B) = = _ n S ( ) B = broccoli spinach,, so n ( B) = n ( B ) P ( B) = = _ n S c P ( ) =, so n ( ) = 3. So, P ( ) = ( ) c ( ) 3 ( S ) n _ = _ n _ n (S) = 3_ = 6 = 1_. 1_ 6. = _ 6. S Image redits: Ted Morrison/Bon ppetit/lamy QUESTIONING STRTEGIES When you calculate the theoretical probabilities of events based on the same probability experiment, can the term in the denominator of the probability ratio change? Explain. No, the term in the denominator corresponds to the sample space, which does not change. If a set has no elements, what is the probability of the event represented by the set? Explain. 0, because there are no outcomes in the sample space that correspond to the event If the elements of a set are the same as the elements of the sample space, what is the probability of the event represented by the set? Explain. 1, because the number of elements in the set is the same as the number of elements in the sample space Module 19 9 Lesson 1 DIFFERENTITE INSTRUTION Manipulatives Encourage students to design their own experiments to illustrate what they have learned about probability, such as calculating the complement of rolling a number with a number cube and then attempting to conform the calculation experimentally. Invite students to demonstrate their experiments before the class. Probability and Set Theory 9

5 EXPLIN Using the omplement of an Event VOID OMMON ERRORS Students may have difficulty understanding when to use the complement to find a probability. Point out that students may be able to find the probability directly but that the complement may provide a shortcut. ontinue to remind students to create Venn diagrams to help them recognize relationships between sets. QUESTIONING STRTEGIES Why are there different equations that relate the probability of an event and its complement? The three equations state the same relationship in different ways. Reflect. Discussion In Example 1B, which is greater, P ( B) or P ( B)? Do you think this result is true in general? Explain. Since P ( B) = 6_ is greater than P ( B) = _, the union is more likely than the intersection. Yes, this is generally true since the union includes all the elements from both events, whereas the intersection contains only elements present in both sets. However, if = B, then the probability of the union and intersection will be the same. Your Turn The numbers 1 through 30 are written on slips of paper that are then placed in a hat. Students draw a slip to determine the order in which they will give an oral report. Event is being one of the first 10 students to give their report. Event B is picking a multiple of 6. If you pick first, calculate each of the indicated probabilities. 3. P () P () = n () n (S) = 10. P ( B) B = 1,, 3,,, 6,, 8, 9, 10, 1, 18,, 30 n ( B) ; P ( B) = n (S). P ( B) B = 6 n ; P ( B) = ( B) = 1 n (S) P ( c ) 30 = 1_ 3 c = 11, 1,, 30 ; P ( c ) = n ( c ) n (S) = 0 30 = _ 3 = 1 30 Explain Using the omplement of an Event You may have noticed in the previous examples that the probability of an event occurring and the probability of the event not occurring (i.e., the probability of the complement of the event) have a sum of 1. This relationship can be useful when it is more convenient to calculate the probability of the complement of an event than it is to calculate the probability of the event. P () + P ( c ) = 1 P () = 1 - P ( c ) P ( c ) = 1 - P () Probabilities of an Event and Its omplement The sum of the probability of an event and the probability of its complement is 1. The probability of an event is 1 minus the probability of its complement. The probability of the complement of an event is 1 minus the probability of the event. Module Lesson 1 LNGUGE SUPPORT onnect Vocabulary Have students create a set of cards with diagrams to help them become familiar with the vocabulary introduced in this lesson. Help students connect the vocabulary to the notation used to represent a set, an element, the universal set, a subset, union, intersection, and complement. Have students use different colors to highlight and distinguish each relationship. 93 Lesson 19.1

6 Example Use the complement to calculate the indicated probabilities. You roll a blue number cube and a white number cube at the same time. What is the probability that you do not roll doubles? Step 1 Define the events. Let be that you do not roll doubles and c that you do roll doubles. Step Make a diagram. two-way table is one helpful way to identify all the possible outcomes in the sample space. White Number ube Blue Number ube , 1 1, 1, 3 1, 1, 1, 6, 1,, 3,,, 6 3 3, 1 3, 3, 3 3, 3, 3, 6, 1,, 3,,, 6, 1,, 3,,, 6 6 6, 1 6, 6, 3 6, 6, 6, 6 Step 3 Determine P ( c ). Since there are fewer outcomes for rolling doubles, it is more convenient to determine the probability of rolling doubles, which is P ( c ). To determine n ( c ), draw a loop around the outcomes in the table that correspond to c and then calculate P ( c ). n ( P ( c ) c ) = _ n (S) = _ 6 36 = 1_ 6 Step Determine P (). Use the relationship between the probability of an event and its complement to determine P (). P () = 1 - P ( c ) = 1-1_ 6 = _ 6 So, the probability of not rolling doubles is 6. One pile of cards contains the numbers through 6 in red hearts. second pile of cards contains the numbers through 8 in black spades. Each pile of cards has been randomly shuffled. If one card from each pile is chosen at the same time, what is the probability that the sum will be less than 1? Step 1 Define the events. Let be the event that the sum is less than 1 and c be the event that the sum is not less than 1. Step Make a diagram. omplete the table to show all the outcomes in the sample space. Step 3 Determine P ( c ). ircle the outcomes in the table that correspond to c, then determine P ( c ). c n 6 P ( c ) = _ = _ n S Black Spades Red Hearts Module 19 9 Lesson 1 Probability and Set Theory 9

7 ELBORTE VOID OMMON ERRORS Students may have trouble identifying some outcomes associated with an event. Encourage students to carefully identify all outcomes by using tables, lists, or diagrams. They can circle the outcomes of interest (often called the favorable outcomes) in the sample space. QUESTIONING STRTEGIES How does listing the elements in a set help you find the probability of an event associated with the set? The probability is based on the number of elements in the set, so you can just count the elements for the numerator of the probability ratio. Step Determine P (). Use the relationship between the probability of an event and its complement to determine P ( c ). P () = - ( ) = - _ 6 = _ 19 1 P c 1 So, the probability that the sum of the two cards is less than 1 is _ 19. Reflect. Describe a different way to calculate the probability that the sum of the two cards will be less than 1. Use the table to count the number of outcomes in event instead of c, which is 19, then divide that by the total number of outcomes to get 19. Your Turn One bag of marbles contains two red, one yellow, one green, and one blue marble. nother bag contains one marble of each of the same four colors. One marble from each bag is chosen at the same time. Use the complement to calculate the indicated probabilities. 8. Probability of selecting two different colors is selecting the same color: (R 1, R), (R, R), (Y, Y), (G, G), (B, B) ; P ( c ) = 0 = 1_, so P () = 1-1_ = 3_. 9. Probability of not selecting a yellow marble is selecting at least one yellow marble: (Y, R), (Y, Y), (Y, G), (Y, B), (R 1, Y), (R, Y), (G, Y),(B, Y) P ( c ) = 8 0 = _, so P () = 1 - _ = 3_. Elaborate SUMMRIZE THE LESSON How can you use set theory to help you calculate theoretical probabilities? You can use the number of elements in a set to define theoretical probability: the theoretical probability n () that an event will occur is given by P () = _ n (S), where S is the sample space. 10. an a subset of contain elements of? Why or why not? No. The elements of a subset are contained completely within the parent set, whereas none of the elements of the complement of a set are in set by definition, and thus they cannot be in a subset of. 11. For any set, what does Ø equal? What does Ø equal? Explain. The intersection of set and the empty set is the empty set, Ø = Ø, since the two sets do not have any elements in common. The union of set and the empty set is set, Ø =, since the elements of the union are the elements in set or the empty set. 1. Essential Question heck-in How do the terms set, element, and universal set correlate to the terms used to calculate theoretical probability? Possible answer: To calculate probability, you need to know the number of possible outcomes in the sample space, which is the number of elements in the universal set. You also need to know the number of possible outcomes in the defined event, which is the number of elements in the defined set. Module 19 9 Lesson 1 9 Lesson 19.1

8 Evaluate: Homework and Practice Set is the set of factors of 1, set B is the set of even natural numbers less than 13, set is the set of odd natural numbers less than 13, and set D is the set of even natural numbers less than. The universal set for these questions is the set of natural numbers less than 13. So, = 1,, 3,, 6, 1, B =,, 6, 8, 10, 1, = 1, 3,,, 9, 11, D =,, 6, and U = 1,, 3,,, 6,, 8, 9, 10, 11, 1. nswer each question. 1. Is D? Explain why or why not.. Is B? Explain why or why not. Yes, because every element of D is also an element of. You have a set of 10 cards numbered 1 to 10. You choose a card at random. Event is choosing a number less than. Event B is choosing an odd number. alculate the probability. 9. P () 10. P (B) The sample space S = 1,, 3,,, 6,, 8, 9, 10 ; Online Homework Hints and Help Extra Practice No, because there is at least one element of B that is not an element of. For example, 8 is an element of B that is not an element of. 3. What is B?. What is?,, 6, 1 1, 3. What is B? 6. What is? 1,, 3,, 6, 8, 10, 1 1,, 3,,, 6,, 9, 11, 1. What is? 8. What is B?,, 9, 10, 11 1, 3,,, 9, 11 The sample space S = 1,, 3,,, 6,, 8, 9, 10 ; = 1,, 3,,, 6 B = P () = _ n () 6_ 3_ 1, 3,,, 9 n (S) = 10 = P (B) = _ n (B) _ 1_ n (S) = 10 = 11. P ( B) 1. P ( B) The sample space S = 1,, 3,,, 6,, 8, 9, 10 ; The sample space S = 1,, 3,,, 6,, 8, 9, 10 ; B = 1,, 3,,, 6,, 9 B = P ( B) = _ n ( B) = 8 1, 3, n (S) 10 = P ( B) = _ n ( B) = 3_ n (S) P ( ) 1. P ( B ) The sample space S = 1,, 3,,, 6,, 8, 9, 10 ; =, 8, 9, 10 n ( ) P ( ) = _ = n (S) _ 10 = _ The sample space S = 1,, 3,,, 6,, 8, 9, 10 ; B =,, 6, 8, 10 n (B ) P (B ) = _ = _ 1_ n (S) 10 = EVLUTE SSIGNMENT GUIDE oncepts and Skills Explore Working with Sets Example 1 alculating Theoretical Probabilities Example Using the omplement of an Event OMMUNITING MTH Practice Exercises 1 8 Exercises 9 1,, 6, 9 Exercises 1 1, 3, 8 Discuss the importance of understanding the sample space. Encourage students to always list the members of the sample space before they find a probability. Discuss why this can help avoid errors, such as finding the probability of rolling a with a number cube as 1. INTEGRTE MTHEMTIL PRTIES Focus on Modeling MP. Discuss when a Venn diagram might be useful in solving a probability problem, and when another method might be easier. Module Lesson 1 Exercise Depth of Knowledge (D.O.K.) Mathematical Practices Recall of Information MP. Modeling Recall of Information MP. Reasoning 1 0 Skills/oncepts MP. Reasoning 1 3 Strategic Thinking MP.6 Precision 3 Strategic Thinking MP. Modeling Strategic Thinking MP.3 Logic 3 Strategic Thinking MP.3 Logic Probability and Set Theory 96

9 INTEGRTE MTHEMTIL PRTIES Focus on Math onnections MP.1 Review the connection between likelihood and probability with students. Discuss how this can be useful when solving problems. When students calculate the probability of an event, be sure they understand what this means in the context of the original problem. For example, students should recognize that an event with a probability of 0.9 is very likely to occur, while an event with a probability of 0.1 is unlikely to occur. VOID OMMON ERRORS Students may not consider the sample space when finding probabilities. Suggest that they summarize the probability ratio using words before they compute the probability. Image redits: Oleg Golovnev/Shutterstock Use the complement of the event to find the probability. 1. You roll a 6-sided number cube. What is the probability that you do not roll a? The probability of rolling a, P (), is 1_ 6. The probability of not rolling a is 1 - P () = = You choose a card at random from a standard deck of cards. What is the probability that you do not choose a red king? The probability of drawing a red king, P (red king) _, is 1_ = 6. The probability of not drawing a red king is 1 - P (red king) = = You spin the spinner shown. The spinner is divided into 1 equal sectors. What is the probability of not spinning a? _ The probability of spinning a, P (), is 1. The probability of not spinning a is 1 - P () = 1-1_ 11_ 1 = bag contains red, blue, and 3 green balls. ball is chosen at random. What is the probability of not choosing a red ball? The probability of choosing a red ball, P (red ball) _ 1_, is 10 =. The probability of not choosing a red ball is 1 - P (red ball) 1 = 1 - =. 19. ards numbered 1 1 are placed in a bag. ball is chosen at random. What is the probability of not choosing a number less than? The probability of choosing a number less than, P (less than ) _ 1_, is 1 = 3. The probability of not choosing a number less than is 1 - P (less than ) 1 = 1-3 = Slips of paper numbered 1 0 are folded and placed into a hat, and then a slip of paper is drawn at random. What is the probability the slip drawn has a number which is not a multiple of or? Multiples of up to 0:, 8, 16, 0 Multiples of up to 0:, 10, 1, 0 The set of multiples of or is,, 8, 10, 1, 16, 0. P (multiple of or ) = _ 0 The probability of not selecting a card that is a multiple of or is 1 - P (multiple of or ) = 1 - _ 13_ 0 = 0. Module 19 9 Lesson 1 Exercise Depth of Knowledge (D.O.K.) Mathematical Practices 8 3 Strategic Thinking MP.3 Logic 9 3 Strategic Thinking MP.3 Logic 9 Lesson 19.1

10 1. You are going to roll two number cubes, a white number cube and a red number cube, and find the sum of the two numbers that come up. a. What is the probability that the sum will be 6? There are 36 possible outcomes. There are ways to get a sum of 6, where the first addend is from the white cube and the second addend is from the red cube: + 1, +, 3 + 3, +, and 1 +. So the probability of getting a sum of 6, P (6), is 36. b. What is the probability that the sum will not be 6? The probability that the sum will not be 6 is P (not 6) = 1 - P (6) = 1-36 = VISUL UES When students create Venn diagrams to model a sample space and sets, caution them to be sure that an element is not used more than once on the diagram. For example, have students check that a number does not appear both in Set and in its intersection with Set B.. You have cards with the letters, B,, D, E, F, G, H, I, J, K, L, M, N, O, P. Event U is choosing the cards, B, or D. Event V is choosing a vowel. Event W is choosing a letter in the word PPLE. Find P (U V W). U V W = ; P (U V W) = 1 16 standard deck of cards has 13 cards (, 3,,, 6,, 8, 9, 10, jack, queen, king, ace) in each of suits (hearts, clubs, diamonds, spades). The hearts and diamonds cards are red. The clubs and spades cards are black. nswer each question. 3. You choose a card from a standard deck of cards at random. What is the probability that you do not choose an ace? Explain. 1 ; there are aces in the -card deck, so P (ace) = 13 = 1. This means 13 P (not ace) = = You choose a card from a standard deck of cards at random. What is the probability that you do not choose a club? Explain. 3_ P (not club) = 1-1_ = 3_. ; there are 13 clubs in the -card deck, so P (club) = 13. You choose a card from a standard deck of cards at random. Event is choosing a red card. Event B is choosing an even number. Event is choosing a black card. Find P ( B ). Explain. = 1_. This means B = Ø because you can never draw a card that is both red and black. Therefore, P ( B ) = 0. Image redits: Sung-Il Kim/orbis Module Lesson 1 Probability and Set Theory 98

11 JOURNL Have students write and solve their own probability problems. Remind students to use set notation in their solutions to the problems. 6. You are selecting a card at random from a standard deck of cards. Match each event with the correct probability. Indicate a match by writing the letter of the event on the line in front of the corresponding probability.. Picking a card that is both red and a heart. B 1_ B. Picking a card that is both a heart and an ace. 1_. Picking a card that is not both a heart and an ace. 1_ = 13 n (red heart) P (red heart) = n (deck) n (heart ace) P (heart ace) = = 1 n (deck) = 1_ P (not (heart ace) ) = 1 - P (heart ace) = 1-1 = 1 ; the only card that is both a heart and an ace is the ace of hearts, so there are 1 cards in the event not (heart ace). H.O.T. Focus on Higher Order Thinking. ritique Reasoning bag contains white tiles, black tiles, and gray tiles. Someone is going to choose a tile at random. P (W), the probability of choosing a white tile, is 1. student claims that the probability of choosing a black tile, P (B), is 3 since P (B) = 1 - P (W) = 1-1 = 3. Do you agree? Explain. No; choosing a black tile is not the complement of choosing a white tile since the bag also contains gray tiles. It is not possible to calculate P (B) from the given information. 8. ommunicate Mathematical Ideas bag contains red marbles and 10 blue marbles. You are going to choose a marble at random. Event is choosing a red marble. Event B is choosing a blue marble. What is P ( B)? Explain. 0; B = Ø since a marble cannot be both red and blue. So P ( B) = ritical Thinking Jeffery states that for a sample space S where all outcomes are equally likely, 0 P () 1 for any subset of S. reate an argument that will justify his statement or state a counterexample. ssume is a subset of S. Then 0 n () n (S). For example, if S has 10 elements, the number of elements of is greater than or equal to 0 and less than or equal to 10. No subset of S can have fewer than 0 elements or more than 10 elements. So 0 _ n () 1. When all the outcomes are equally likely, n (S) P () = _ n (). Therefore 0 P () 1. n (S) Module Lesson 1 99 Lesson 19.1

12 Lesson Performance Task For the sets you ve worked with in this lesson, membership in a set is binary: Either something belongs to the set or it doesn t. For instance, is an element of the set of odd numbers, but 6 isn t In 196, Lofti Zadeh developed the idea of fuzzy sets to deal with sets for which membership is not binary. He defined a degree of membership that can vary from 0 to 1. For instance, a membership function m L (w) for the set L of large dogs where the degree of membership m is determined by the weight w of a dog might be defined as follows: dog is a full member of the set L if it weighs 80 pounds or more. This can be written as m L (w) = 1 for w 80. dog is not a member of the set L if it weighs 60 pounds or less. This can be written as m L (w) = 0 for w 60. dog is a partial member of the set L if it weighs between 60 and 80 pounds. This can be written as 0 < m L (w) < 1 for 60 < w < 80. Small Dogs The large dogs portion of the graph shown displays the m membership criteria listed above. Note that the graph shows 1 only values of m(w) that are positive. 1. Using the graph, give the approximate weights for which a dog is considered a full member, a partial member, and not a member of the set S of small dogs. 0 pounds to 0 pounds; between 0 pounds and 30 pounds; more than 30 pounds. The union of two fuzzy sets and B is given by the membership rule m B (x) = maximum (m (x), m B (x) ). So, for a dog of a given size, the degree of its membership in the set of small or medium-sized dogs (S M) is the greater of its degree of membership in the set of small dogs and its degree of membership in the set of medium-sized dogs. The intersection of and B is given by the membership rule m B (x) = minimum (m (x), m B (x)). So, for a dog of a given size, the degree of its membership in the set of dogs that are both small and medium-sized (S M) is the lesser of its degree of membership in the set of small dogs and its degree of membership in the set of medium-sized dogs. Using the graph above and letting S be the set of small dogs, M be the set of medium-sized dogs, and L be the set of large dogs, draw the graph of each set. a. S M b. M L Degree of Membership m pproximately size shown here Weight (lb) w Degree of Membership m Degree of Membership 0 0 Medium-Sized Dogs pproximately size shown here Weight (lb) w Large Dogs Weight (lb) Module Lesson 1 w INTEGRTE MTHEMTIL PRTIES Focus on Reasoning MP. all attention to the point in the Lesson Performance Task graph where the green mediumsized dog line and the blue big-dog line intersect. sk students to give as much information as they can about that point. Sample answer: The point represents a weight of around 100 pounds and a degree of membership of around 0.3. The point represents the highest degree of membership that a dog of around 100 pounds can obtain simultaneously in both the medium-sized and big-weight categories. INTEGRTE MTHEMTIL PRTIES Focus on ritical Thinking MP.3 rf has a small-dog degree of membership of x and a medium-sized dog degree of membership of y. Is x > y, x < y, or does the relationship between x and y depend on rf s weight? Explain. The relationship depends on rf s weight. The red and green graphs intersect at about 0 pounds. If rf weighs less than 0 pounds, x > y. If rf weighs more than 0 pounds, x < y. If rf weighs 0 pounds, x = y. EXTENSION TIVITY Have students draw graphs showing fuzzy sets ranging from cold to hot. (Three sets could show cold, warm, and hot. Four sets could show cold, cool, warm, and hot. However, leave the choice of adjectives and the number of sets to students.) The vertical axis should record degrees of membership from 0 to 1. The horizontal axis should show either Fahrenheit or elsius temperatures. Encourage students to be creative with their graphs, for example, by using colors to distinguish sets from one another. Students should write and answer at least three questions involving unions, intersections, and complements of the sets they have graphed. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Probability and Set Theory 960