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Copyright © 2010 Pearson Education, Inc. Unit 3: Gathering Data Chapter 11 Understanding Randomness
Copyright © 2010 Pearson Education, Inc. 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 © 2010 Pearson Education, Inc. 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 © 2010 Pearson Education, Inc. 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 © 2010 Pearson Education, Inc. It’s Not Easy Being Random 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 © 2010 Pearson Education, Inc. 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 © 2010 Pearson Education, Inc. Practical Randomness We need an imitation of a real process so we can manipulate and control it. In short, we are going to simulate reality. Suppose cereal manufacturers put famous athlete cards in boxes to boost sales. 20% of the boxes contain Tiger Woods, 30% David Beckham, and 50% Serena Williams. You want all 3 cards. How many boxes do you expect to buy to get the complete set?
Copyright © 2010 Pearson Education, Inc. A Simulation The sequence of events we want to investigate is called a trial. The basic building block of a simulation is called a component. Trials usually involve several components. In the card example, a component would be opening one cereal box. After the trial, we record what happened—our response variable. Trial’s outcome - # of components (boxes) in the sequence There are seven steps to a simulation…
Copyright © 2010 Pearson Education, Inc. Simulation Steps 1.Identify the component to be repeated. Opening a box of cereal 2.Explain how you will model the component’s outcome. Digits from 0 to 9 are equally likely to occur. 0 and 1 represent Woods (2 digits out of 10 = 20%). 2, 3, and 4 represent Beckham. 5, 6, 7, 8, and 9 represent Williams. 3.Explain how you will combine the components to model a trial. Pretend to open boxes until the collection is complete. Look at each random digit and indicate what picture it represents. 4.State clearly what the response variable is. We want to find the # of boxes it might take to get all 3 cards. 5.Run several trials. 6.Collect and summarize the results of all the trials. Report the shape, center, and spread 7.State your conclusion. In context!
Copyright © 2010 Pearson Education, Inc. Just Checking – p. 260 The baseball World Series consists of up to 7 games. The first team to win 4 games wins the series. The first two are played at one team’s home ballpark, the next three are at the other team’s park, and the final two are played back at the first park. Records over the past century show that there is a home field advantage; the home team has about 55% chance of winning. Does the current system of alternating ballparks even out the home field advantage? How often will the team that begins at home win the series? Set up a simulation: 1. What is the component to be repeated?
Copyright © 2010 Pearson Education, Inc. Just Checking Continued… 2. How will you model each component from equally likely random digits? 3. How will you model a trial by combining components? 4. What is the response variable? 5. How will you analyze the response variable?
Copyright © 2010 Pearson Education, Inc. Generating Random Numbers on Calculator Math PRB 5:randInt( Examples of randInt simulations: randInt(0,1) randomly chooses a 0 or a 1 Commonly used for coin tosses randInt(1,6) Produces a random integer from 1 to 6 Commonly used for rolling a die randInt(1,6,2) Simulates rolling two dice. Hit enter to do several rolls in a row.
Copyright © 2010 Pearson Education, Inc. Generating Random Numbers on Calculator Examples of randInt simulations (continued): randInt(0,9,5) Produces 5 random integers that might represent the pictures on the cereal box (0-9 had different meanings) randInt(0,56,3) Produces three random integers from 0 to 56. Also know how to use Appendix G in book when technology isn't available.
Copyright © 2010 Pearson Education, Inc. Example: Suppose a basketball player has an 80% free throw success rate. How can we use random numbers to simulate whether or not she makes a foul shot? How many shots might she be able to make in a row without missing? Describe the waiting time simulation of having her shoot free throws until she misses, counting the number of successes. Component: Trial: Response variable: Statistic:
Copyright © 2010 Pearson Education, Inc. 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. Run enough trials. Simulation is cheap and fairly easy to do.
Copyright © 2010 Pearson Education, Inc. What have we learned? How to harness the power of randomness. A simulation model can help us investigate a question when we can’t (or don’t want to) collect data, and a mathematical answer is hard to calculate. How to base our simulation on random values generated by a computer, generated by a randomizing device, or found on the Internet. Simulations can provide us with useful insights about the real world.
Copyright © 2010 Pearson Education, Inc. Chapter 11 Assignments: pp. 265 – 267 Day 1: # 1, 2, 4, 5, 7 – 10,13,15,17 Day 2: # 11, 12, 19, 25, 28, 29, 31 – 33, 37
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide
Copyright © 2010 Pearson Education, Inc. Slide A small town employs 34 salaried, nonunion employees. Each employee receives an annual salary increase.
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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.
Chapter 11 Understanding Randomness At the end of this chapter, you should be able to Identify a random event. Describe the properties of random.
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Randomness Has structure in the long run Randomness seems “Fair” 1) Nobody can predict the outcome ahead of time. 2) Some underlying set of outcomes are.
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