Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 11- 1

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 11 Understanding Randomness

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Why Be Random? What is it about chance outcomes being random that makes random selection seem fair? Two things: Nobody can guess the outcome before it happens. When we want things to be fair, usually some underlying set of outcomes will be equally likely (although in many games some combinations of outcomes are more likely than others).

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Why Be Random? (cont.) Example: Pick “heads” or “tails.” Flip a fair coin. Does the outcome match your choice? Did you know before flipping the coin whether or not it would match?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Why Be Random? (cont.) Statisticians don’t think of randomness as the annoying tendency of things to be unpredictable or haphazard. Statisticians use randomness as a tool. But, truly random values are surprisingly hard to get…

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide It’s Not Easy Being Random

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide It’s Not Easy Being Random (cont.) It’s surprisingly difficult to generate random values even when they’re equally likely. Computers have become a popular way to generate random numbers. Even though they often do much better than humans, computers can’t generate truly random numbers either. Since computers follow programs, the “random” numbers we get from computers are really pseudorandom. Fortunately, pseudorandom values are good enough for most purposes.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide It’s Not Easy Being Random (cont.) There are ways to generate random numbers so that they are both equally likely and truly random. The best ways we know to generate data that give a fair and accurate picture of the world rely on randomness, and the ways in which we draw conclusions from those data depend on the randomness, too.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A Simulation A simulation consists of a collection of things that happen at random. The most basic event is called a component of the simulation. Each component has a set of possible outcomes, one of which will occur at random.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A Simulation (cont.) The sequence of events we want to investigate is called a trial. Trials usually involve several components. After the trial, we record what happened—our response variable. There are seven steps to a simulation…

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Simulation Steps 1.Identify the component to be repeated. 2.Explain how you will model the outcome. 3.Explain how you will simulate the trial. 4.State clearly what the response variable is. 5.Run several trials. 6.Analyze the response variable. 7.State your conclusion (in the context of the problem, as always).

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example 1 If some one handed you a coin and told you that its biased towards usually landing heads, you’d be justly skeptical. What would you do?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More Questions How many times would you need to flip the coin? What would be the behavior of a fair coin?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Even more things to ponder If we tossed the coin 100 times what would be the number of heads that would make you believe the coin was biased?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Random generator Calculator randint() What numbers could we use to represent heads and tails?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Run Test How will we count all the ones and zeros?

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Statistics That branch of mathematics in which we never know exactly what we are talking about or whether anything we say is true

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Can Go Wrong? Don’t overstate your case. Always be sure to indicate that future results will not match your simulated results exactly. Model the outcome chances accurately. Run enough trials.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What have we learned? We will harness the power of randomness. A simulation model can help us investigate a question for which: many outcomes are possible, we can’t (or don’t want to) collect data, and a mathematical answer is hard to calculate. We base our simulations on random values. Like all models, simulations can provide us with useful insights about the real world.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Day 2 Chapter 11 Random – Without bias Bias – Systematic favoring of certain outcomes Population – Entire group of individuals that we want information about Sample – Part of the population that we actually examine in order to gather information Simple Random Sample (SRS) – Selection method that ensures every possible sample has an equal chance of being the sample ultimately selected Simulation – Imitation of chance behavior based on a model that accurately reflects the experiment under consideration

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Ways to generate random data: Computer Calculator Random digit table

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A Random Digit Table A table of random digits is a long string of the digits 0,1,2,3,4,5,6,7,8,9 with these two properties: 1. Each entry in the table is equally likely to be any of the 10 digits 0 through The entries are independent of each other. That is, knowledge of the one part of the table gives no information about the other part.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Random Digit Table

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide How to Design a Simulation Using a Probability Sample Space: 1. Identify what is to be repeated. 2. Identify the response variable. 3. Explain how to model the outcomes and run the simulation. (Detailed directions!) 4. Identify how many trials.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Simulation Example: A popular kid’s cereal box contains either a Lance Armstrong card (30% probability), a Serena Williams card (50% probability), or a rarer Tiger Woods card (20% probability). Explain how one could simulate the outcomes if you wanted to collect all three of card types.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Answers 1. Identify what is to be repeated: the selections of a cereal box. 2. Identify the response variable: What are we interested on? We want to know how many boxes it take to get all three pictures. This is the response variable.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Problem continued 3. Explain how to model the outcomes and run the simulation. (Detailed directions!) The digits from 0 to 9 are equally likely to occur. Because 20% of the boxes contain Tiger’s picture, we will use 2 of the ten digits to represent that outcome. Three of the 10 digits can model 30% of the boxes with Lance Armstrong card, and the remaining 5 digits can represents the 50% of the boxes with Serena Williams. One possible assignment of the digits is 0,1 = Woods 2,3,4 = Armstrong 5,6,7,8,9= Williams

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Problem continued Explain how will you simulate the trial: A trail is the sequence of events that we are pretending will take place. In this case we want to pretend to open cereal boxes until we have one of each picture. We do this by looking at each random number and indicating what outcome it represents. We continue until we’ve encountered all three pictures.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Problem continued For example, the random number would mean you get Lance’s picture (2) in the first box you open, Serena’s picture (9) in the next box, tow more Armstrong picture (2,4) in the next row boxes, and then the Tiger woods picture (0) you need to complete the collection. Since we have gotten all three pictures, we are finished with one trail of our simulation. Therefore in the sample trial here, the response variable is 5 boxes

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Run Several Trials A simulation is cheaper that really buying cereal and the more trials you run the better. Look at first row of table. Let’s figure out how many boxes of cereal need to be opened to get all three pictures.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Make a table to keep all data organized. Trial #outcomeY=# boxes

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Problem Continued Analyze The response variable: We wanted to know how many boxes we might expect to buy to get all three card. To answer the question, we need to analyze the response variable. We find the average number of boxes State you conclusion ( in the context of the problem, as always) Bases on our simulation, we estimate that customers who want the complete set of sport star pictures will buy in average ___ boxes of cereal.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Your Turn Work with a partner and complete Simulating Randomness Activity

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Practice Problem A company with a large sales staff announces openings for three positions as regional managers. Twenty-two of the current salespersons apply, 12 men and 10 women. After the interview, when the company announces the newly appointed managers, all three go to women. The men complain of job discrimination. Do they have a case? Simulate a random selection of three people from the applicant pool and make a decision about the likelihood that a fair process would result in hiring all women.

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Problem 2 A certain college estimates that the 3-score SAT total for students who apply for admission can be described by a Normal model with a mean of 1570 and a standard deviation of 180. Admissions officers search the pile of envelopes. Opening them at random to look for three applicants with SAT totals over How many envelopes do you think they will need to open.