1. Chapter 11 – Understanding Randomness 1. What is a random event? Nobody can guess the outcome before it happens. Let’s try an experiment. On the next page are four numbers. Pick one of the numbers at random and write it down. Don’t shown anyone.

2. Chapter 11 – Understanding Randomness Pick one of the numbers at random and write it down. Don’t shown anyone. I chose number ________.

3. August 24 th, 2015 Results

4. Chapter 11 – Understanding Randomness 5% of all people pick the number 1 75% of all people pick the number 3 20% of all people pick either 2 or 4 Humans do not do well at just picking something at random.

5. Chapter 11 – Vocabulary 3. Pseudorandom number – produced by a computer when the computer’s random number generator is used Computers cannot produce truly random numbers. Instead they use a random number table.

6. Chapter 11 – Understanding Randomness

7. Chapter 11 – Calculator Skills Random Integer MATH Choose ‘PRB’ tab Choose ‘5: randInt(‘ 4. RANDINT(Low, High, How many) RANDINT(1,100,5) will produce five random integers between 1 and 100 inclusive.

8. Chapter 11 – Calculator Skills 5. Every TI-84 will produce the same pseudorandom integers if the calculator starts at the same place in the calculator’s random number table. To “seed” the calculator, means to tell the calculator where to start on it’s random number table.

9. Chapter 11 – Calculator Skills Seed the Random Number Generator 1) Enter a number 2) STO> 3) MATH Choose ‘PRB’ tab Choose ‘1: rand‘ “Seed”: 0  rand

10. Chapter 11 – Calculator Skills Seed the Random Number Generator Results are: 95 91 15 52 41 “Seed”: 0  rand Generate 5 random numbers 1 to 100: RANDINT(1, 100, 5) These are only Pseudorandom numbers. Because the outcomes were predictable.

11. Time for a test Put your name at the top of a sheet of paper and number it from 1 to 30. Each question is multiple choice and there are 5 choices…. A, B, C, D, E GO! What? You need the questions? Why? Admit it you didn’t study… so just guess. You have 30 seconds.

12. Time for a test Time is up. I trust you. Grade your own work. How did you do? Do you know the probability that you passed? That’s nice, but it doesn’t matter if you do or don’t yet. (That’s a later chapter.) But we will look at a method to figure it out in this chapter. 16. B

13. Chapter 11 – Vocabulary 6. Simulation – consists of a sequence of random outcomes that model a situation

14. Chapter 11 – Understanding Randomness 7. The Steps for Conducting a Simulation 1. Identify the component to be repeated. 2. Explain how you will model the Outcome. 3. Explain how you will simulate the trial. A component is the most basic event in a simulation The possible results of a component, one of which will occur at random, are called outcomes. The sequence of events being investigated (usually involves multiple components) are called a trial.

15. Chapter 11 – Understanding Randomness Steps for Conducting a Simulation 1. Identify the component to be repeated. 2. Explain how you will model the outcome. 3. Explain how you will simulate the trial. 4. Clearly state the response variable. 5. Run several trials. 6. Analyze the response variable. 7. State your conclusion in context. The result of interest from a trial is called the response variable.

16. Chapter 11 – Understanding Randomness 8. Thirty-eight percent of the people in the United States have type O positive blood. Three unrelated people in the U.S. are selected at random. Design a simulation to determine how often all three will have type O positive blood.

17. Chapter 11 – Understanding Randomness 8. Thirty-eight percent of the people in the United States have type O positive blood. Three unrelated people in the U.S. are selected at random. Design a simulation to determine how often all three will have type O positive blood. a. Identify the component to be repeated. Component: Check the blood type of 1 person.

18. Chapter 11 – Understanding Randomness 8. Thirty-eight percent of the people in the United States have type O positive blood. Three unrelated people in the U.S. are selected at random. Design a simulation to determine how often all three will have type O positive blood. a. Identify the component to be repeated. Component: Check the blood type of 1 person. b. Explain how you will model the outcome. Let 00 – 37 represent the person selected has type O positive blood. Let 38 – 99 represent the person selected doesn’t have type O positive blood.

19. Chapter 11 – Understanding Randomness 8. Thirty-eight percent of the people in the United States have type O positive blood. Three unrelated people in the U.S. are selected at random. Design a simulation to determine how often all three will have type O positive blood. c. Explain how you will simulate the trial. Each trial consists of identifying 3 pair of digits as Y(Person has O positive) or N(Person Doesn’t have O positive). Each pair represents one person. We want 3 persons. d. Clearly state the response variable. The response variable is whether or not all three persons have type O positive blood type.

20. 8. Thirty-eight percent of the people in the United States have type O positive blood. Three unrelated people in the U.S. are selected at random. Design a simulation to determine how often all three will have type O positive blood. e. Run several trials. We’ll run 8 trials using the following random numbers: Trial #Outcomes All O+? 1 2 3 4 :: ::: 30/Y 73/N 47/NNO 15/Y71/N 83/N NO 72/N 27/Y 97/N NO 12/Y25/Y 77/N NO 7 29/Y 28/Y 31/Y YES 5 NO 6 NO 8 YES Y Y Y Y Y N N N N N N N

21. Chapter 11 – Understanding Randomness 8. Thirty-eight percent of the people in the United States have type O positive blood. Three unrelated people in the U.S. are selected at random. Design a simulation to determine how often all three will have type O positive blood. f. Analyze the response variable. 2 / 8 or 25% of the trials found all three were O+ g. State your Conclusion in context. Our simulation showed that if three unrelated people in the U.S. were chosen at random, all three are O+ about 25% of the time. However, it should be noted that only 8 trials were run.

22. Chapter 11 – Understanding Randomness 1. Identify the component to be repeated. 2. Explain how you will model the outcome. 3. Explain how you will simulate the trial. 4. Clearly state the response variable. 5. Run several trials. 6. Analyze the response variable. 7. State your conclusion in context. REVIEW THE The Steps for Conducting a Simulation

23. Chapter 11 – Understanding Randomness 9. Thirty-eight people out of 100 in the United States have type O positive blood. Design a simulation to determine how many people on average would need to be selected at random to get one person that has type O positive blood.

24. Chapter 11 – Understanding Randomness 9. Thirty-eight people out of 100 in the United States have type O positive blood. Design a simulation to determine how many people on average would need to be selected at random to get one person that has type O positive blood. a. Identify the component to be repeated. Component: Check the blood type of 1 person b. Explain how you will model the outcome. Let 01 – 38 represent that the person selected has type O positive blood. Let 39 – 99 & 00 represent that the person selected doesn’t have type O positive blood.

25. Chapter 11 – Understanding Randomness 9. Thirty-eight people out of 100 in the United States have type O positive blood. Design a simulation to determine how many people on average would need to be selected at random to get one person that has type O positive blood. c. Explain how you will simulate the trial. Each trial consists of identifying a pair of digits as N(Person doesn’t have O positive) or Y(Person has O positive). Each pair represent one person. We will keep checking until a person has been identified as having O+ blood. d. Clearly state the Response variable. The response variable is how many individuals were checked to find the first person with type O positive blood.

26. 9. Thirty-eight people out of 100 in the United States have type O positive blood. Design a simulation to determine how many people on average would need to be selected at random to get one person that has type O positive blood. e. Run several trials. We’ll run 12 trials using the following random numbers: Trial #Outcomes # of People 1 2 3 4 5 95/N 83/N462/N25/Y 30/Y 1 91/N 78/N58/N02/Y 4 10/Y 1 34/Y 1

27. Chapter 11 – Understanding Randomness 9. Thirty-eight people out of 100 in the United States have type O positive blood. Design a simulation to determine how many people on average would need to be selected at random to get one person that has type O positive blood. f. Analyze the response variable. 4+1+4+1+1+1+1+1+1+1+4+4 = 24 24/12 = 2 g. State your Conclusion in context. Our simulation showed that if unrelated people in the U.S. were chosen at random that on average 2 people would need to be selected to get a person with O+ blood. However, it should be noted that only 12 trials were run.

28. Chapter 11 – Understanding Randomness 1. Identify the component to be repeated. 2. Explain how you will model the outcome. 3. Explain how you will simulate the trial. 4. Clearly state the response variable. 5. Run several trials. 6. Analyze the response variable. 7. State your conclusion in context. REVIEW THE The Steps for Conducting a Simulation

29. Chapter 11 – Understanding Randomness 10. Design a simulation to determine on average how many times someone should roll a pair of dice in order to get “snake eyes” (Two ones). a. Identify the Component to be repeated. Component: Rolling a pair of dice once

30. The chance of rolling “snake eyes” isn’t the same as the chance of rolling a seven. Fill-in the table below to see this. ONETWOTHREEFOURFIVESIX ONE TWO THREE FOUR FIVE SIX 2 34567 3 4 5 6 7 8 4 5 6 7 8 9 5 6 7 8 9 10 6 7 8 9 10 11 7 8 9 10 11 12 2: 1/363: 2/364: 3/365: 4/366: 5/36 7: 6/368: 5/369: 4/3610: 3/3611: 2/36 12: 1/36

31. Chapter 11 – Understanding Randomness 10. Design a simulation to determine on average how many times someone should roll a pair of dice in order to get “snake eyes” (Two ones). b. Explain how you will model the outcome. Let 00 represent rolling “snake eyes”. Let 01 – 35 represent not rolling “snake eyes”. Ignore 36 – 99. If a pair for this set is found check the next pair of digits.

32. Chapter 11 – Understanding Randomness 10. Design a simulation to determine on average how many times someone should roll a pair of dice in order to get “snake eyes” (Two ones). c. Explain how you will simulate the trial. Each trial consists of identifying a pair of digits as N(Not “snake eyes”) or Y(“snake eyes”) or I(ignore). Each pair represents one roll of two dice. If a pair of digit is to be ignored, then they will not be counted as a roll of the dice. The trail will end once “snake eyes” is rolled. d. Clearly state the response variable. The response variable is how many times the pair of dice were rolled in order to get “snake eyes”.

33. Chapter 11 – Understanding Randomness 10. Design a simulation to determine on average how many times someone could roll a pair of dice in order to get “snake eyes” (Two ones). e. Run several trials. We’ll run 2 trials using the following random numbers: T#T# OutcomesCtCt 1 2 I NINI INIIN IN IIN I N I I I I Y 8 65358 70469 87149 89509 72176 18103 55169 79954 72002 20582 05409 20831 01911 60767 55248 79253 00317 84120 77772 50103 75642 64510 79185 86109 67056 01991 14620 23598 88515 35696 N N I N I I N N N I N N I I N I I I Y 10

34. Chapter 11 – Understanding Randomness 10. Design a simulation to determine on average how many times someone could roll a pair of dice in order to get “snake eyes” (Two ones). f. Analyze the response variable. 8 + 10 = 18 18/2 = 9 h. State your conclusion in context. From our simulation, on average we expect to roll the pair of dice nine times in order to get “snake eyes”. However, it should be noted that only 2 trials were run.

35. Chapter 11 – Understanding Randomness 11. How often will someone who completely guesses on a 29 question exam pass the exam? a. Identify the Component to be repeated. Component: Answering one test question b. Explain how you will model the outcome. Since there is a 20% chance of guessing correctly… Let 0 – 1 represent guessing correctly Let 2 – 9 represent not guessing correctly

36. Chapter 11 – Understanding Randomness 11. How often will someone who completely guesses on a 29 question exam pass the exam? c. Explain how you will simulate the trial. Each trial consists of identifying a digit as R( guessed right) or W(guessed wrong). Each digit represents one question. There are 29 questions, so stop after checking 29 digits. d. Clearly state the response variable. Since 21 right out of 29 is passing, the response variable is were 21 questions answered correctly?

37. Chapter 11 – Understanding Randomness 11. How often will someone who completely guesses on a 29 question exam pass the exam? e. Run several trials. You’ve been given a random number table. How many trails can you run? f. Analyze the response variable. How many times did a trial end up with someone passing? g. State your conclusion in context. What would your conclusion be?