CC-15. This sounds like a great idea for a television game show! MATHEMATICAL. Lesson Vocabulary conditional probability contingency table

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1 1 Interactive earning Solve It! State Standards MA.9.S-P.2.6 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A... Also MA.9.S-P.1.1, MA.9.S-P.1.3, MA.9.S-P.1.4, MA.9.S-P.1.5, MA.9.S-P.2.8 MP 1, MP 2, MP 3, MP 4, MP 6 PURPOSE To analyze a conditional probability Objectives To find conditional probabilities To use tables and tree diagrams to determine conditional probabilities situation PROESS Students may act out the situation. draw a branching tree diagram. FAIITATE Q What is the probability that you originally guessed the correct box? In this case, what wins the prize? [ 13 ; not switching boxes] This sounds like a great idea for a television game show! Q What is the probability that you did not guess the correct box? In this case, what wins the prize? [ 23; switching boxes] Q How does the probability of winning by switching compare to the probability of winning by not switching? [The probability of winning by A great prize is inside Box 1, 2, or 3. You choose Box 1. You are shown that Box 2 contains a rubber chicken. You are given a chance to change your choice. Should you stay with Box 1 or should you change to Box 3? Justify your answer. MATHEMATIA esson Vocabulary conditional probability contingency table ASWER See Solve It in Answers on next page. OET THE MATH In the Solve It, students Essential Understanding onditional probability exists when two events You write the conditional probability of event B, given that event A occurs, as P (B A). You read P (B A) as the probability of event B, given event A. A contingency table, or two-way frequency table, is a frequency table that contains data from two different categories. ontingency tables and tree diagrams can help you find conditional probabilities. Problem 1 Finding A What is P (female graduate school)? Problem 1 What s the condition? The student is at a graduate school. Q Multiply the probabilities of selecting a female and selecting a graduate student. Is the product the same as the probability in 1A? Explain. [o; it is The condition that the person selected is at graduate school limits the sample space to the 3,303,000 graduate students. Of those, 1,9,000 are female. 19 P (female graduate school) = hapter 11 A conditional probability is the probability that one event occurs, given that another event has occurred. Math Background onditional probability is the probability that one event will occur given that another event has occurred. For two events A and B, the conditional probability that B will occur, given that A has occurred, is written P (B 0 A). Once it is known that an event A has occurred, the sample space is restricted to the subset of events that include A. For example, if you are rolling a number cube, then the original sample space is 51, 2, 3, 4, 5, 66. If A is the event that an even number is rolled, and you know that A occurs, then the new sample space is 52, 4, 66. If B is the event that a 2 or 4 is rolled, then P (B) = 13. However, P (B 0 A) = 23. s (in thousands) s (in thousands) Two-year colleges Four-year colleges Graduate schools SOURE: U.S. ensus Bureau Probability and Statistics Essential Understanding Student Genders Education The table shows students by gender and by type of school in You pick a student at random. 2 Guided Instruction Big idea Probability What s th The waste be recycled are dependent. calculate conditional probability based on an initial choice. In the lesson, students will find conditional probability of situations using formulas and tree diagrams. Preparing to Teach 2 3 PRATIES The probability that an event, B, will occur given that another event, A, has already occurred is called a conditional probability. switching is twice that of not switching.] much lower, about Selecting a grad student is a given in 1A, so that part of the probability is 100%.] 1 Recall that if two events are independent, then the occurrence of HSM15_A2Reg_SE_TrKit.indd Page 03/08/13 3:53 PM gg-018 one event does not affect the probability that the second event will occur. Thus, if and D are independent events, then P (D 0 ) = P (D). Mathematical Practices Attend to precision. Students will use a clear definition of the term conditional probability and determine when to calculate it. HSM15_A2Reg_SE 15_TrKit.ind 02/08/13 3:04 PM /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/A...

2 Problem 1 B What is P (female)? ty total number of females Q In 1B, is 0. a reasonable answer? Explain. [Yes; P (F) = total number of students = the number of females is greater than the number of males in every row of the table, so the probability of picking a female must be greater than %.] 9933 = 17, a. In Problem 1, what is P (Four@year male)? b. Reasoning Without calculating, given a student is enrolled in a four-year college, is it more likely for the student to be male or female? Explain. Problem 2 in Statistics What s the condition? The waste sample has to be recycled waste. Q How would you translate 1a into a question using only words? [What is the probability that a Municipal Waste ollected (millions of tons) Multiple hoice Americans recycle increasing amounts through municipal waste collection. The table shows the collection data for What is the probability that a sample of recycled waste is paper? Material Recycled Paper Metal % 33% Glass 28% % Plastic Other The given condition is that the waste is recycled. A favorable outcome is that the recycled waste is paper. student attends a four-year college, given that the student is male?] ot Recycled Problem 2 Q Based on the answer, how would you find the probability that a sample of recycled waste is not paper without calculating? [The probability SOURE: U.S. Environmental Protection Agency of recycled waste that is not paper is the probability of all the recycled waste minus the probability of recycled waste that is paper: 100% % = 43%.] 45.2 P (paper recycled) = , or % The probability that the recycled waste is paper is about %. The correct answer is D. 2. a. What is the probability that a sample of recycled waste is plastic? b. What is the probability that a sample of recycled waste is glass? Q Does the question What is the probability that a sample of waste is recycled glass? differ from the question in 2b? Explain. [Yes; in 2b, it is a given You can use a formula to find conditional probability. that the sample is recycled.] s) Key oncept Take ote For any two events A and B with P (A) 0, P(A and B) P(B A) = P (A) Q In the formula for conditional probability, is A or B the given condition? [A is the given condition.] Q Why must P (A) not equal zero? [Division by zero is not allowed.] esson 11-4 Page 55 02/08/13 9: PM gg /08/13 3:04 PM 55 onditional Probability /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/A Answers HSM15_A2Reg_SE_TrKit.indd Page 55 03/08/13 3:53 PM gg-018 /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/A... Solve It! hange to box 3; there is a 13 chance that box 1 has the prize and a 23 chance that box 3 has the prize. 1. a or.355% b. female; there are more females enrolled. 2. a or 2.64% b or 4.03% 55

3 Using the formula for conditional probability, you can calculate a conditional probability from other probabilities. Problem 3 Another approach to Problem 3 is to recognize that once it is given that the customer is male, the sample space is restricted to the first row of the table. To find the size of the new sample space, add the entries in the first column, + 8 = 20. This is the total number of males in the sample space. The number of males who pay online is. Dividing, you get 20 = 0.6. Problem 3 Using the Formula Market Research A utility company asked of its customers whether they pay their bills online or by mail. What is the probability that a customer pays the bill online, given that the customer is male? To use P (online male) Q Suppose you know that a customer pays by mail. Is = it more likely that the customer is male or female? Explain. [14 customers pay by mail. Of these, Online By Mail What are branches point? The tree fir at gradua not gradu of these br happy a happy. 20 P (male) = P (male and online) = P (male and online), P (male) you need P(male and online) and P(male). eight are male and six are female. Therefore, the 8 probability that a customer is male is 14? 0.. Thus, it is more likely that a customer is male, given that the customer pays by mail, even though there are more females in the original sample space.] Bill Payment P (online male) = There are 20 males, and males who pay online out of total customers. Substitute and simplify. P (male and online) P (male) = 20 = 3 = = The probability that a customer pays online given that the customer is male is 0.6. Q What two probabilities do you need to find to 3. Researchers asked shampoo users solve this problem? [P(female and directly onto head) and P(female)] whether they apply shampoo directly to the head, or indirectly using a hand. What is the probability that a respondent applies shampoo directly to the head, given that the respondent is female? Q Without using the formula, how would you find P (Directly on Head 0 )? [Divide the number in the female and directly onto head box by the total in the female row.] It follows from P (B A) = P (A and B) P (A) Which pa you follo Follow the represents who are h their prese Applying Shampoo that P (A and B) = P (A) Directly Onto Head Into Hand First # P (B A). You can use this rule along with a tree diagram to find probabilities of dependent events. hapter 11 Probability and Statistics HSM15_A2Reg_SE 15_TrKit.ind Additional Problems Answers HSM15_A2Reg_SE_TrKit.indd Page 03/08/13 3:53 PM gg The table shows the number of male and female freshmen who chose to play one of the three intramural sports offered at a small college. Basketball 40 Soccer Volleyball 10 What is P (soccer 0 female)? ASWER a bout A student compiled the following table comparing the amounts of land area and water area, in square miles, in various U.S. states. and Water Alaska 1,936 91,332 Florida,018 11,777 Texas 262, alifornia 1, What is the probability that a point chosen at random on a map is water, given that the map is of Florida? ASWER a bout 17.9% 3. The table shows the number of male and female customers at a café on a certain day. Each customer drank either a soda or an iced tea. /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/A... Soda Iced Tea What is the probability that a customer drank a soda, given that the customer was male? ASWER The chance of rain for the next evening is 60%. If it rains, the chance of lightning is 80%. If it does not rain, the chance of lighting is 5%. What is the probability that it will not rain and there will be no lightning? ASWER 3 8% (continued)

4 Problem 4 Using a Tree Diagram Problem 4 Education A school system compiled the following information from a survey it sent to people who were juniors 10 years earlier. A tree diagram is an alternative way to represent a sample space. The advantage of a tree diagram is that you can mark different probabilities for each branch; in an ordered-list sample space the probability of each item in the set is assumed to be equal and it is the number of combinations of items that determines probability. 85% of the students graduated from high school. Of the students who graduated from high school, 90% are happy with their present jobs. Of the students who did not graduate from high school, 60% are happy with their present jobs. What are the branches at each point? The tree first branches at graduated and not graduated. Each of these branches at happy and not happy. What is the probability that a person from the junior class 10 years ago graduated from high school and is happy with his or her present job? Make a tree diagram to help organize the information. et G = graduated, G = not graduated, H = happy with present job, and H = not happy with present job. This branch represents the students who did graduate. G Each first branch represents a simple probability P(G) 0.85 and P(G) G 0.90 H 0.10 H 0.60 H Q The given information in the problem states the probability a student graduated high school. How was the probability that a student did not graduate found? [Every student either graduated or did not graduate, so the total probability must be 1, = 0.15.] Each second branch represents a conditional probability P(H G) 0.90 and P(H G) 0.40 Q The tree diagram shows four paths. What do you think the sum of the probabilities for all four paths should be? Explain and verify your answer. [The sum of the probabilities for all 0.40 H four paths represents everything that could happen, so the probability should be 100%. P (H and G) + P (H and G) + P (H and G) + P (H and G) = = 1.] This branch represents the students who did not graduate. Which path should you follow? Follow the path that represents graduates who are happy with their present job. The blue highlighted path represents P(G and H). P (G and H) = P (G) = 0.85 # P (H G) # 0.90 = Q Which of the four paths did you follow to answer The probability that a person from the junior class 10 years ago graduated and is happy with his or her present job is 0.765, or 76.5%. this question? What is the probability of each section of the path? [The path is not graduated 4. What is the probability that a student from the junior class 10 years ago in and is happy. P (G) = 0.15 and P (H G) = 0.6] Problem 4 did not graduate and is happy with his or her present job? Q If the graduating class contained 600 students, how many students would you expect did not graduate and are happy with their present jobs? [ students] esson % HSM15_A2Reg_SE_TrKit.indd Page 03/08/13 3:53 PM gg-018 /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/A...

5 3 esson heck Do you know HOW? For Exercises 1 3, students must understand that it is given that the drawn card is black. That fact does not need to be calculated. Therefore, the denominator for calculating these probabilities should be 26 rather than 52. For Exercise 4, students may benefit from drawing a tree diagram. They should determine that the first branch is for illustrated/notillustrated and the second branches are for hardback/not-hardback. esson heck Do you know HOW? 2. P (4) 3. P(diamond) 6. Open-Ended Describe a situation in which you would use conditional probability to find the answer. 4. The probability that a car has two doors, given that it is red, is 0.6. The probability that a car has two doors and is red is 0.2. What is the probability that a car is red? Practice Do you UDERSTAD? App ha MATHEMATIA PRATIES See Problems 1 and 2. Use the table to find each probability. 8. P(has diploma) In Exercise 5, students may think that the sum of the probabilities of paired branches should equal the probability of the preceding branch in the diagram. Explain that the smaller branches contain 100% (all) of the preceding branch, which is the same as a part of the entire diagram. For Exercise 6, the situation must have at least two choices or events, and the first should be given. B 7. ompare and ontrast How are the Fundamental ounting Principle and tree diagrams alike? How are they different? Practice and Problem-Solving Exercises A PRATIES 5. Reasoning Using the tree diagram in Problem 4, explain why the probabilities on each pair of branches must add up to 1. A card is drawn from a standard deck of cards. Find each probability, given that the card drawn is black. 1. P(club) MATHEMATIA Do you UDERSTAD? haracteristics of Job Applicants 9. P(has diploma and experience) Has Experience 10. P(has experience has diploma) Has High School Diploma 11. P(has no diploma has experience) Yes o Yes o Use the table to find each probability.. P(The recipient is male.) Projected umber of Degree Recipients in 2010 (thousands) 13. P(The degree is a bachelor s.) 14. P(The recipient is female, given that the degree is an associate s.) lose 15. P(The degree is not an associate s, given that the recipient is male.) Q What is a simple probability experiment that shows how the probability of an outcome changes when a part is given? [Answers may vary. Sample: Degree Associate s Bachelor s SOURE: U.S. ational enter for Education Statistics See Problems 3 and 4. Use the survey results for Exercises 16 and Find the probability that a respondent has a pet, given that the respondent has had a pet. Flipping a coin twice and getting two heads has a probability of one fourth. If the first coin flip is given as heads, then the probability of getting two heads is one half. The probability doubled.] 39% have a pet now and have had a pet. 17. Find the probability that a respondent has never had a pet, given that the respondent does not have a pet now. 61% do not have a pet now. 86% have had a pet. 14% do not have a pet now and have never had a pet. 18. Sports A football team has a 70% chance of winning when it doesn t snow, but only a 40% chance of winning when it snows. Suppose there is a % chance of snow. Make a tree diagram to find the probability that the team will win. 58 hapter 11 Probability and Statistics 58 ommon58ore HSM15_A2Reg_SE 15_TrKit.ind Answers but not with the Fundamental ounting Principle, you can determine probabilities of dependent events, or conditional probabilities. HSM15_A2Reg_SE_TrKit.indd Page 58 8/13/13 :29 AM epg /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/S... esson heck , or about 7.7% 3. 0% 4. % 5. The sum of the probability of an event happening and the probability of an event not happening is 1. Each branch represents either the event happening or the event not happening. 6. heck students work. 7. Answers may vary. Sample: Tree diagrams apply to cases in which more than one event occurs in a sequence. The Fundamental ounting Principle applies to situations in which there are multiple outcomes of a single event. With a tree diagram, 58 Practice and Problem-Solving Exercises % 0. 4 snow no s now 0. 3 P(W) = 55% % W W W win loss

6 19. Make a tree diagram based on the survey results below. Then find P(a female respondent is left-handed) and P(a respondent is both male and right-handed). Of all the respondents, 17% are male. Of the male respondents, 33% are left-handed. Of female respondents, 90% are right-handed. B Apply ASSIGMET GUIDE Basic: 8 19 all, Suppose A and B are independent events, with P (A) = 0.60 and P (B) = Find each probability. a. P (A and B) b. P (A B) c. What do you notice about P (A) and P(A B)? d. Reasoning One way to describe A and B as independent events is The occurrence of B has no effect on the probability of A. Explain how the answer to part (c) illustrates this relationship. r. e Average: 9 19 odd, Advanced: 9 19 odd, Mathematical Practices are supported by exercises with red headings. Here are the Practices supported in this lesson: 21. Think About a Plan A math teacher gives her class two tests. 60% of the class passes both tests and 80% of the class passes the first test. What percent of those who pass the first test also pass the second test? What conditional probability are you looking for? How can a tree diagram help you solve this problem? nd 2. nd 4. 4 Practice MP 1: Make Sense of Problems Ex. 21 MP 2: Reason Quantitatively Ex. 32a MP 3: onstruct Arguments Ex. 5, 6 MP 3: ommunicate Ex. 5, 20d, 32a MP 3: ompare Arguments Ex. 7 MP 4: Model with Mathematics Ex. 33 Weather Use probability notation to describe the chance of each event. et S,, W, and R represent sunny, cloudy, windy, and rainy weather, respectively. 22. cloudy weather 23. sunny and windy weather 24. rainy weather if it is windy Applications exercises have blue headings. Exercises 18 and 25 support MP 4: Model. 25. Transportation You can take Bus 65 or Bus 79. You take the first bus that arrives. The probability that Bus 65 arrives first is 75%. There is a 40% chance that Bus 65 picks up passengers along the way. There is a 60% chance that Bus 79 picks up passengers. Your bus picked up passengers. What is the probability that it was Bus 65? The tree diagram relates snowfall and school closings. Find each probability. et H,, O, and represent heavy snowfall, light snowfall, schools open, and schools closed, respectively. hallenge et. 26. P() 27. P(H and O) 28. P (H ) 29. P ( O) 30. P ( ) 31. P (H O) M R F 0.9 R 0.17 M F R HSM15_A2Reg_SE_TrKit.indd Page 03/08/13 3:53 PM gg-018 male female right-handed left-handed P ( 0 F ) = 10%, P(M and R) 11.4% 20. a b c d. Since P (A) = P (A 0 B), the probability of A is the same, regardless of the occurrence of B % 22. P ( ) 23. P (S and W ) 24. P (R 0 W ) 2, or 66.67% 3 0., or % 0.08, or 8% O O H 0.4 To check students understanding of key skills and concepts, go over Exercises 9, 17, 20, 21, and a. Writing Explain which branches of the tree diagram at the right represent A conditional probabilities. Give a specific example. b. Are the event of having a license and the event of being an adult independent events? Justify your answer. M c. Open-Ended Estimate probabilities for each branch of the tree diagram A adult (21 or older) for your city or town. Then find P(). M minor (under 21) 33. Reasoning Sixty percent of a company s sales representatives have licensed driver completed training seminars. Of these, 80% have had increased sales. not licensed to drive Overall, % of the representatives (whether trained or not) have had increased sales. Use a tree diagram to find the probability of increased sales, given that a representative has not been trained. esson HOMEWORK QUIK HEK 32. a. The branches labelled and represent conditional probabilities dependent upon the person being an adult or a minor. For example, the top branch represents the probability that a person is licensed given that he or she is an adult. b. o; the probability of a minor being licensed is not the same as the probability of an adult being licensed. c. heck students work. 8 I 33. T 0. /0/PE014/TRASITIO_KITS/A/AIARY/2015/XXXXXXXXXX/ayout/Interior_Files/A I R 0. 8 T representative that completed training seminars R representative that didn t complete a training seminar I representative with increased sales representative without increased sales P (I 0 ) = 0.2