Simulating Probability of Compound Events Learning Goal In this lesson, you will: Use simulations to estimate compound probabilities. Do you know that there are four major blood groups: A, B, O, and AB. Did you know that the percent of people having each blood group differs by race and by country? For example, in some countries nearly everyone has the same blood group, and in other countries there are people who have each of the 4 blood groups. In the United States, the percent of people having each blood group differs by race and ethnicity. You may have noticed that there are often blood drives sponsored by the American Red Cross. These drives encourage people to donate blood that others can use if they are critically injured in accidents or natural disasters, or have serious diseases. 17.4 Simulating Probability of Compound Events 939
Problem 1 Students are given data involving the percents of the population having group A, group B, group O, and group AB blood. They will determine various probabilities of a person or persons having a specified blood group. Using a random number table, a random digit generator on a calculator, or a computer spreadsheet, students will model the problem, run simulations consisting of 20 trials, and use the data to answer related questions. The results should be close to the theoretical probabilities. Grouping Have students complete Questions 1 through 3, part (c) with a partner. Then share the responses as a class. Share Phase, Questions 1 and 2 What is the probability that the next person who enters the Community Center to donate blood does not have group A blood? Problem 1 Blood Groups Overall, the percent of people in the U.S. having each blood group is given in the table. The percents have been rounded to the nearest whole number percent. Blood Groups A B O AB Percent of Population 42% 10% 44% 4% Suppose the Red Cross is having a blood drive at the Community Center. 1. What is the probability that the next person who enters the Community Center to donate blood has group A blood? P(group A) 5 0.42 The probability of the next person to donate group A blood is 42%. 2. What is the probability that the next person who enters the Community Center to donate blood has group A or group O blood? P(group A or group O) 5 P(A) 1 P(0) 5 0.42 1 0.44 5 0.86 The probability of the next person to donate group A or group O blood is 86%. The first two questions involved events that the probability could be determined by using basic knowledge of probability. However, many events involve more advanced rules of probability. In most cases, though, a simulation can be used to model an event. 940 Chapter 17 Probability of Compound Events
Share Phase, Question 3, parts (a) through (c) What is the probability that the next person who enters the Community Center to donate blood has group O blood? How many different blood groups are there? How many different groups will be needed to assign numbers to people? What is a way two digit numbers could be assigned to people? 3. Determine the probability that out of the next 5 people to donate blood, at least 1 person has group AB blood. a. What could be a good model for simulating people who donate blood? Answers will vary depending on your students responses. The answers provided should be used as sample answers. b. You will use a random number table, a random digit generator on a calculator, or computer spreadsheet to model the problem. How could you assign numbers to people to account for the different blood types? I could assign the numbers 00 through 41 to people with group A blood. I could then assign the numbers 42 through 51 to people with group B blood. Next, I could assign the numbers 52 through 95 to people with group O blood. Finally, I could assign 96 through 99 to people with group AB blood. c. Describe one trial of the simulation. One trial could consist of choosing five 2-digit numbers from the random digit table, and counting how many of the 2-digit numbers are 96 through 99. Grouping Have students complete Questions 3, part (d) through 4 with a partner. Then share the responses as a class. d. Conduct 20 trials of the simulation and record your results in the table. 17.4 Simulating Probability of Compound Events 941
Trial Number Numbers Count of Numbers from 96 through 99 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 942 Chapter 17 Probability of Compound Events
Share Phase, Questions 3, part (d) through 4 How does your table of 20 trials compare to your classmates tables? Is your probability that out of the next 5 people to donate blood, at least one of them has group AB blood close to the theoretical probability? Explain. How does your table of 20 trials compare to your classmates tables? Is your probability for the number of people expected to donate blood before a person with group B blood enters close to the theoretical probability? Explain. e. Out of the 20 trials, how many had at least 1 number from 96 through 99? f. According to your simulation, what is the probability that out of the next 5 people to donate blood, at least one of them has type AB blood? The theoretical probability is, 0.185. 4. How many people would be expected to donate blood before a person with group B blood would donate blood? a. Describe one trial of the simulation. One trial would consist of choosing 2-digit random numbers until a number from 42 through 51 is chosen. b. Conduct 20 trials of the simulation and record your results in the table. 17.4 Simulating Probability of Compound Events 943
Trial Number Number of 2-digit numbers chosen until a number from 42 through 51 appears 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 944 Chapter 17 Probability of Compound Events
Problem 2 A basketball player typically makes 5 out of 6 free throws. Students design a model for simulating the player shooting free throws. They then conduct 20 trials and use the results to determine the experimental probability of the player making all 3 free throws. Students also design and conduct a simulation to model the number of times the player would shoot before missing a shot. Grouping Have students complete Questions 1 through 4 with a partner. Then share the responses as a class. c. Calculate the average for your 20 trials. d. About how many people would be expected to donate blood before a person with group B blood enters? Answers should be close to 10. Problem 2 Shoot Out The star player on the basketball team typically makes 5 out of 6 free throws. There is a foul and the player gets to shoot 3 free throws. 1. What might be a good model for simulating the player shooting free throws? A 6-sided number cube 2. How could you assign the numbers on the cube to model the player shooting free throws? I could assign the numbers 1 through 5 to represent the player making a shot, and the number 6 to represent the player missing a shot. Share Phase, Questions 1 through 3 How can a number cube be used to model this situation? How many sides are on the number cube? What numbers on the cube represent the player making the shot? What numbers on the cube represent the player missing the shot? What is another way to simulate a trial? 3. What is the probability that the player makes all 3 free throws? a. Describe one trial of the simulation. One trial would consist of rolling a number cube 3 times, and recording how many times the cube showed a number 1 through 5. b. Conduct 20 trials of the simulation and record your results in the table. 17.4 Simulating Probability of Compound Events 945
Trial Number Number of times a 1 through 5 occurs 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 946 Chapter 17 Probability of Compound Events
c. Count the number of times that the result of all three numbers are between 1 and 5. d. According to your simulation, what is the probability the player makes all 3 free throws? The theoretical probability is, 0.58. Share Phase, Question 4 What is the fastest way to simulate a trial? Do 20 trials result in enough data to see the experimental probabilities approach the theoretical probabilities? Explain. Could a number cube also be used to model the number of times the player would shoot before missing a shot? Explain. 4. Design and conduct a simulation to model the number of times the player would shoot before missing a shot. Example: I could us a number cube. I could let 1 through 5 represent the player making the shot, and let the number 6 represent the player missing the shot. A trial consists of rolling the number cube until I roll a 6 and counting how many times it takes to roll the 6. I would repeat the trial a number of times and determine the mean. The theoretical answer is 1 out of 6. Be prepared to share your solutions and methods. 17.4 Simulating Probability of Compound Events 947