2. Copyright © 2009 Pearson Education, Inc. Chapter 11 Understanding Randomness

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: Accurately model a situation through simulation. Discuss the results of a simulation study and draw conclusions about the questions being investigated.

4. Copyright © 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 © 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 © 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 © 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 © 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

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 Practical Randomness Motivating example: A cereal manufacturer puts pictures of famous athletes on cards in boxes of cereal in hopes of boosting sales. 20% of the boxes contain a picture of Tiger Woods, 30% a picture of David Beckham, and the rest a picture of Serena Williams. If you want all three pictures, how many boxes of cereal do you expect to have to buy? We need an imitation of a real process so we can manipulate and control it. In short, we are going to simulate reality.

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 A Simulation We want to understand the typical number of boxes we’ll have to buy and how that number varies (and the shape of the distribution) so we have to test this many times. The sequence of events we want to investigate is called a trial. (e.g. opening boxes until we get all three cards) The basic building block of a simulation is called a component. (e.g. opening one box) There are seven steps to a simulation…

11. Copyright © 2009 Pearson Education, Inc. Slide 1- 11 Simulation Steps 1.Identify the component to be repeated. 2.Explain how you will model the component’s outcome using random numbers. 3.State clearly what the response variable is. 4.Explain how you will combine the components into a trial to model the response variable. 5.Run several trials. 6.Collect and summarize the results of all the trials. 7.State your conclusion.

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 Example: Simulation Steps 1.Identify the component to be repeated. Opening a box of cereal 2.Explain how you will model the component’s outcome using random numbers. Digits 0-9 are equally likely to occur, use 0,1 to indicate Tiger, 2,3,4 to indicate Beckham, and 5, 6, 7, 8, 9 to indicate Serena 3.State clearly what the response variable is. Number of boxes it took to get all three pictures 4.Explain how you will combine the components into a trial to model the response variable. We open boxes (repeat components) until all three pictures found. Trials outcome is number of boxes (components) 5.Run several trials. 6.Collect and summarize the results of all the trials. Be sure to report shape, center, and spread 7.State your conclusion. The simulation suggests that…

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 Example: Simulation Steps Use 0,1 to indicate Tiger, 2,3,4 to indicate Beckham, and 5, 6, 7, 8, 9 to indicate Serena 1.Run several trials using the following random digits: 221772630438741009253708627058199 76227258497959070328250011089633217535 82264380029225464494376064238904376655 72 2.Fill out a table: Trial number, component outcomes, Trial outcomes: y = number of boxes) 3.Collect and summarize the results of all the trials. Be sure to report shape, center, and spread 4.State your conclusion.

14. Copyright © 2009 Pearson Education, Inc. Slide 1- 14 What Can Go Wrong? Don’t overstate your case. Beware of confusing what really happens with what a simulation suggests might happen. Model outcome chances accurately. A common mistake in constructing a simulation is to adopt a strategy that may appear to produce the right kind of results. Are all outcomes equally likely? Run enough trials. Simulation is cheap and fairly easy to do.

15. Copyright © 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 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- 15

16. Copyright © 2009 Pearson Education, Inc. Slide 1- 16 examples You decide to play the lottery. You have to pick 5 numbers between 1 and 60 and want to use random numbers to pick your lucky numbers. Which numbers would you play based on these random digits: 43680 98750 13092 76561 58712 Is this a particularly good or bad strategy? Text #11, 13, 15

17. Copyright © 2009 Pearson Education, Inc. Bad simulations: Explain why these fail to model the real situation properly Use a random number from 0 through 9 to represent the number of heads that appear when 9 coins are tossed A basketball player takes a foul shot. Look at a random digit, using an odd digit to represent a good shot and an even digit to represent a miss. Use five random digits from 1 through 13 to represent the denominations of the cards in a poker hand. Wrong conclusion: A Statistics student properly simulated the length of a checkout lines in a grocery store and then reported, “The average length of the line will be 3.2 people.” What is wrong with this conclusion? Slide 1- 17

18. Copyright © 2009 Pearson Education, Inc. More Practice You are pretty sure that your candidate for class president has about 55% of the votes in the entire school. But you’re worried that only 100 students will show up to vote. How often will the underdog (the one with 45% support) win? Describe how you will simulate a component and its outcomes Describe how you will simulate a trial Describe the response variable Slide 1- 18

19. Copyright © 2009 Pearson Education, Inc. Slide 1- 19 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.

20. Copyright © 2009 Pearson Education, Inc. Slide 1- 20 Example – Actually conduct this simulation using all 7 steps (#31 in your textbook) 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.

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 Example – Actually conduct this simulation using all 7 steps (#33 in your textbook) 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?

22. Copyright © 2009 Pearson Education, Inc. Slide 1- 22 Hand-in homework You are taking a multiple choice quiz which consists of 6 questions. Each question has five possible answers to choose from of which only one is correct. Answer the following questions. 1. (0.25 points) What is your chance of choosing the correct answer for any particular problem? (express as a decimal rounded to two decimal places.) 2. (0.25 points) Given the set of random numbers 0,1,2,...9, explain how you would do a simulation for the above quiz for question #1. 3. (1 points) Actually do a simulation on the TI for all 6 questions. Show the result and interpret the result in the context of the problem. ie which did you get right and which did you get wrong.