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Geometry Items to Support Formative Assessment Unit 5: Probability of Compound Events Part I: Probability of Compound Events Understand independence and conditional probability and use them to interpret data. S.CP.A.1 Describe events as subsets of 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"). S.CP.A.1 Item 1: 1. Far Flight High School has 162 students enrolled in French, 294 students enrolled in Spanish, and 78 students taking both languages. The remaining 132 students do not take a language. Find each of the probabilities below and use appropriate notation to summarize the event. a. b. c. What is the probability that a student takes French or Spanish or both? What is the probability that a student takes French and Spanish? What is the probability that a student does not take French? 2. How would you describe the students you would include if you were considering the complement of P(French Ç Spanish)? What is the probability of this occurring? Solution: 1. a. P(French È Spanish) » 0.74 b. P(French Ç Spanish) » 0.15 c. P(Frenchc) » 0.68 2. The students who are not taking a language would be the complement French The probability is approximately 0.26. Spanish. S.CP.A.1 Item 2: A local coffee shop, Fourbucks, is interested in collecting statistics about its patrons to determine next steps for developing future marketing strategies. The Venn diagram below represents the results of the survey conducted about what patrons purchased during their visit. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Use the information from the Venn diagram to write and answer three questions using the correct notation. Be sure to include at least one intersection, union, and complement. Possible Solutions: If a patron is selected at random, determine the following: a. P(purchasing a drink itemc) = 25 = 0.25 100 95 = 0.95 100 15 c. P(Purchasing a Drink Item Ç Purchasing a Food Item) = = 0.15 100 b. P(Purchasing a Drink Item È Purchasing a Food Item) = S.CP.A.1 Item 3: Write a scenario based on the survey results shown below. Then write 3 probability problems based on your scenario. Possible Solutions: Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. In a sample of 500 randomly selected people, 187 said they owned an mp3 player; 285 said they owned a smartphone; 75 reported they owned both, and 103 said they didn’t own either. P(smartphone È mp3 player) = 0.794 P(owns an mp3 playerc) = 0.626 P(smartphone) = 0.57 S.CP.A.1 Item 4: 250 people in the lodge at Moose Mountain Ski Resort were asked if they were there to ski or snowboard. The results of the survey are shown in the two-way table below. Skis Doesn’t Ski Total Snowboards 95 58 153 Doesn’t Snowboard 60 37 97 Total 155 95 250 If a person at the lodge is selected at random, determine the probability of the events below. 1. P(skis) 2. P(doesn’t participate in either sport) 3. P(snowboards È skis) Solution: 1. P(skis) = 0.24 2. P(doesn’t participate in either sport) = 0.148 3. P(snowboards È skis) = 0.852 S.CP.A.1 Item 5: (Adapted from NCTM Mathematics Assessment Sampler Grades 9-12) Twenty-five people watched a movie showing at the AMC Theater. Of them, fifteen order a drink, ten order popcorn, and nine order candy. Two people order all three items, one orders a drink and candy, three order a drink and popcorn, and one orders popcorn and candy. a. Display the results in a Venn diagram. b. What is the probability that a moviegoer who buys a drink will also buy popcorn. Solution: Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. a. b. 5/15 or 1/3 Use the rules of probability to compute probabilities of compound events in a uniform probability model. S.CP.B.7 Apply the Addition Rule, P(A or B) = P(A)+P(B)-P(A and B), and interpret the answer in terms of the model. S.CP.B.7 Task: The Bar Harbor Boating Company in Maine offers 3 boating tours: Just Whale Sightings, Just Puffin Sightings, and Whale and Puffin Sightings. The manager would like to guarantee the sighting of whales and/or puffins on all of the tours or he would give his customers a full refund, however, he is worried he might lose too much money. If the proportion of sighting only a whale is 0.75, the proportion of sighting only a puffin is 0.95, and the proportion of sighting a whale and puffin is 0.72, set up a two-way table. Then, use what you know about calculating probability to make a recommendation to the manager whether or not he should guarantee sightings for each tour. Include a two-way table to support your recommendation. Sample Answer: Puffins No Puffins Total: Whale 0.72 0.03 0.75 No Whales 0.23 0.02 0.25 Total: 0.95 0.05 1.00 P(Puffin È Whale) = 0.95+0.75-0.72= 0.98 P(Puffin Ç Whale) = 0.72 Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. I would recommend that the manager guarantee the sighting of puffins on the Just a Puffin tour, since the probability of seeing a puffin on a random day is 0.95. I would not recommend the manager offer the guarantee for the sighting of whales on Just the Whale tour, since the probability of seeing a whale on a random day is only 0.75. For the Whale and Puffin tour, I would recommend that the manager offer a guarantee for the sighting of either puffins or whales, since the probability of sighting both a puffin and whale on a random day is only 0.72, but the probability of sighting a puffin or a whale is 0.98. S.CP.B.7 Item 1: At Bayside High School 42% of their students participate in a sport after school, 27% of their students participate in an after school club, and 15% of students participate in both a club and sport after school. The principal, Mr. Belding, would like to determine how involved their students are in after school activities. Create a two-way table to display the data, and find the probability that a randomly selected Bayside student will participate in either a club or sport after school. Answer: Participates in a Sport Does Not Participate in a Sport Total Participates in a Club 0.15 0.12 0.27 Does not participate in a Club 0.27 0.46 0.73 Total 0.42 0.58 1.00 P(Club È Sport): 0.42+0.27-0.15= 0.54 S.CP.B.7 Item 2: Below is the two-way table representing a survey of the proportion of 100 random moviegoers that were interviewed about whether or not they saw the first two movies in The Hunger Games series. Saw The Hunger Games Did not see The Hunger Games Total 0.29 0.04 0.33 Did not see Catching 0.13 0.44 0.67 Saw Catching Fire Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. Fire Total 0.42 0.58 1.00 The CFO of the local movie theatre conducted the interview with his patrons to determine whether or not to show the third movie. To help with his decision, he determined the probability that a randomly selected patron had seen either The Hunger Games or Catching Fire. Below is his work: P(The Hunger Games È Catching Fire) = 0.42 + 0.33 = 0.75 Determine if he is correct and explain why or why not. If he is incorrect, provide the correct probability. Possible Solution: He is incorrect in his calculations, since he forgot to subtract the patrons that saw both The Hunger Games and Catching Fire. If he does not subtract these patrons, then they will count twice. The correct probability is P(The Hunger Games È Catching Fire) = 0.42+0.33–0.29=0.46. S.CP.B.7 Item 3: (From NCTM Mathematics Assessment Sampler Grades 9-12) A warning system installation consists of two independent alarms having probabilities of operating in an emergency of 0.95 and 0.90, respectively. Find the probability that at least one alarm operates in an emergency. Show your work. a. b. c. d. e. 0.995 0.975 0.95 0.90 0.855 (Note: This problem was originally a TIMMS item. It allows students to solve the problem using a variety of approaches, some more complicated than others. The phrase “at least” gives students the opportunity to show their understanding of the complement of the event that neither of the alarms works.) Solution: a Strategy 1: (1 - (0.05)(0.1)) = 1 - 0.005 = 0.995 Other valid student work: Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. S.CP.B.7 Item 4: (Note: This table is also used with an item for S.CP.A.3) The table below represents the distribution of students that attended the Spirit Dance. Freshmen Sophomores Juniors Seniors Attended Spirit Dance 240 185 265 270 Did Not Attend Spirit Dance 60 105 45 30 1. Find the probability that a student selected at random is a junior OR is a student that attended the spirit dance. 2. Find the probability that a student selected at random is a senior OR is a student that did not attend the spirit dance. Solutions: 1. Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. 960 + 45 1005 = 1200 1200 Or P(A or B) = P(A) + P(B) – P(A and B) 310 960 265 1005 + = 1200 1200 1200 1200 2. 240 + 270 510 = 1200 1200 Or P(A or B) = P(A) + P(B) – P(A and B) 300 240 30 510 + = 1200 1200 1200 1200 Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.