1 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Chapter 9 Understanding Randomness

2 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide 1- 2 Objectives: 32. Explain what it means to be random. 33. Accurately model a situation through simulation. 34. Discuss the results of a simulation study and draw conclusions about the questions being investigated.

3 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 9.1 What is Randomness?

4 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Random Procedures A random procedure is a procedure whose outcome cannot be known in advance How can we determine the probability a random procedure will have a certain outcome?

5 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Approach #1: Relative Frequency Approximation of Probability Conduct/observe a procedure n times, and count the # of times that an outcome of interest occurs. Based on these results, the probability of the outcome is estimated as follows: This approach obtains an approximation (estimate) instead of an exact value Law of Large #’s: As the # of trials increases, the relative frequency probability approaches the actual probability

6 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Approach #2: Classical Approach to Probability Assume that a given procedure has n different outcomes and that each of these outcomes has an equal chance of occurring. If an outcome of interest can occur in s of these n ways, then This approach requires equally likely outcomes

7 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Approach #3: Subjective Probabilities P(outcome) is estimated by using personal judgment about the likelihood of an event This approach is needed when there is no repeatable random experiment available Examples: What is the probability it will rain tomorrow? What is the probability the stock market will rise tomorrow? What is the probability more than 5 students in this class will get an “A” on the next exam?

8 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Simulation There are many situations when we want to estimate the probability of an outcome of a random procedure, but: The classical approach to probability is not possible, and/or The relative frequency approach is unwieldy/expensive/infeasible/etc. In these situations we can simulate the random procedure in order to estimate the probability of our outcome of interest We need random numbers to simulate the randomness in real life

9 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Where do we get random numbers? Random number tables from random sources in nature Your textbook has a random number table in Appendix D Computers and your calculator can generate pseudorandom numbers. StatCrunch: Data → Simulate Data Use MATH -> PRB -> randInt(low, high, numtrials) to generate a set of random integers E.g. randInt(0,1,5) will generate a set of 5 random digits of 0 or 1. Slide 1- 9

10 Copyright © 2014, 2012, 2009 Pearson Education, Inc. How to use random numbers It depends on the context… if your outcome occurs with probability 1/5 then you could generate a set of random numbers on your calculator from 1 to 5 and let 1 represent your outcome occurring. RandInt(1,5,100) will generate 100 such digits Or you could use a random number table (digits 0-9) and let 0 and 1 be your outcome occurring Or use RandInt(0,9,100) to generate 100 such digits Or you could use a random number table and look at 2-digit numbers and let 0-19 be your event occurring. Or use RandInt(0,99,100) to generate 100 such numbers There are many ways to do this. Slide 1- 10

11 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Choosing 10 Random Numbers from 1 to 4 with StatCrunch

12 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 9.2 Simulating by Hand

13 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Picking Sports Cards Cereal boxes contain one of three athlete cards: 20% chance of Hope Solo 30% chance of Danica Patrick 50% chance of Blake Griffin How many boxes would you expect to buy to get one of each? It is too expensive to buy hundreds of boxes and open them. Instead, use simulation.

14 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Steps for Simulation Specify how to model a component outcome using equally likely random digits: 1. Identify the component to be repeated. Opening a cereal box 2. Explain how you will model the experiment’s outcome. Assign outcomes to the equally likely random digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 0 or 1 → Hope Solo 2, 3, or 4 → Danica Patrick 5, 6, 7, 8, or 9 → Blake Griffin

15 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Steps for Simulation Specify how to simulate trials: 3. Explain how you will combine the components to model the trial. Look at each digit until all three types are found. 4. State clearly what the response variable is. We want to find the number of boxes it takes to get all three pictures. Put it all together to run the simulation: 5. Run several trials

16 Copyright © 2014, 2012, 2009 Pearson Education, Inc. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Using TI-83/84 RandInt(0,9,10) … we may only need 3 picks, but 10 Will give us more than enough. With StatCrunch: The first column rounded is 6, 5, 7, 3, 2, 1, 3, 8, 5, 0, 0, 2, … It took 6 picks to have at least one of each type of card. The second column rounded is 5, 1, 0, 4, 4, 7, 5, 8, 8, 4, 4, 3, 6, …. It took 4 picks to have at least one of each. Continue this process until you have a large sample.

17 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Steps for Simulation Analyze the response variable: 6. Collect and summarize the results of the trials. As you have learned, look for shape, center, spread, outliers, etc. 7. State your conclusion We estimate it takes a median of 5 boxes to complete the collection, but it could take a lot more. Trial Number Number of Boxes The first four trials organized in a table.

18 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Simulating a Dice Game Rules Roll a total higher than your opponent without going over 21. If your opponent rolled an 18, how many rolls do you expect to make and what is your chance of winning? How will you simulate the components? A component is one roll of the die. Generate numbers from 1 to 6. (RandInt(1,6,10))

19 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Dice Game of 21 How will you combine the components? What is the response variable? I will add up the numbers until I get a total greater than 18 and count the number of rolls. If the total ≤ 21, it is a win. If the total > 21, it is a loss. Use StatCrunch and display a few outcomes: 1, 3, 5, 1, 2, 3, 6, 5, 6, 3, …: 7 tosses. Total: 21 win 2, 2, 4, 1, 5, 6, 1, 1, 4, 5, …: 6 tosses. Total: 20 win 2, 3, 1, 2, 5, 2, 1, 6, 3, 1, …: 8 tosses. Total: 22 loss

20 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Conclusions Suppose you ran 30 trials and tallied the results as shown. What are your conclusions? I typically roll about five or six times for each game. I expect to win about 70% of the time.

21 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Lottery for the Dorms 57 students are in a lottery for the spacious triple dorm room. 20 were from the varsity team and all three winners were from this team. How likely is this? Was it rigged? Plan → Simulation Components: Selection of the students. Outcomes: Generate numbers from 1 to will represent the team members. Trial: Pick the first three distinct numbers. Response Variable: Yes if all three are 1-20

22 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Show Mechanics Use StatCrunch or RandInt to generate 100 sets of three distinct random numbers between 1 and 57. Some are: 5,24,45; 26,56,27; 47,11,51; 35,16,9; 33,15,13; 47,37,32; 21,43,6; 15,22,28; 24,13,42; 38,33,17; 26,36,52; 28,3,55; 25,51,38; 36,18,34; 9,40,52; 5,17,2; 24,49,20; 22,38,45; 38,54,56; Analyze Only 3 out of the 100 trials resulted in “All Varsity.”

23 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Conclusions In the simulation, only 3 out of 100 were “All Varsity.” While 3% is only a small chance, it is not impossible. It looks pretty suspicious. Is 3% a small enough chance to make a formal accusation?

24 Copyright © 2014, 2012, 2009 Pearson Education, Inc. What Can Go Wrong? Don’t Overstate Your Case Simulation is not reality, it only indicates probability. Model Outcome Chances Accurately What would be wrong with generating random numbers 0, 1, 2, 3 to indicate the number of team members? There is not a 25% chance of each. They are not equally likely. Run Enough Trials Don’t just do a few trials. Err on the side of a large number of trials.

25 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide Example – Actually conduct this simulation using all 7 steps Many couples want to have both a boy and girl. If they decide to continue to have children until they have one child of each sex, what would the average family size be? Assume that boys are girls are equally likely.

26 Copyright © 2014, 2012, 2009 Pearson Education, Inc. Slide Example – Actually conduct this simulation using all 7 steps You are playing a children’s game in which the number of spaces you will get to move is determined by the rolling of a die. You must land exactly on the final space in order to win. If you are 10 spaces away, how many turns might it take you to win?