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CHAPTER 5 Probability: What Are the Chances? 5.1: Randomness, Probability, and Simulation Complete 5.1 HW #1 tonight The Practice of Statistics, 5th Edition Starnes, Tabor, Yates, Moore Bedford Freeman Worth Publishers Do Now Activity As a special promotion for its 20-ounce bottle of soda, a soft drink printed a message on the inside of each bottle cap. Some of the caps said, “Please try again!” while others said, “You’re a winner!” The company advertised the promotion with the slogan “1 in 6 wins a prize.” The prize is a free 20-ounce bottle of soda. Seven friends each buy one 20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. The store owner is concerned about losing money from giving away too many free sodas. She wonders if this group of friends is just lucky or if the company’s 1-in-6 claim is inaccurate. The Practice of Statistics, 5th Edition 2 Do Now Activity In this activity, you will perform a simulation to help answer this question. We can model the status of an individual bottle with a six-sided die: let 1-5 represent “Please try again!” and 6 represent “You’re a winner!” 1. Roll your die seven times to imitate the process of the seven friends buying their sodas. How many of them won a prize? 2. Come up and record the data for a class dotplot. 3. Everyone repeat until we have 40 repetitions of the simulation. The Practice of Statistics, 5th Edition 3 Dotplot and results 0 1 2 3 4 5 6 7 Number of winners in a group of seven 4. Discuss results. What percent of the time did the friends come away with three or more prizes, just by chance? Does it seem plausible that the company is telling the truth, but that the seven friends just got lucky? Explain. The Practice of Statistics, 5th Edition 4 The Idea of Probability Chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. Probability describes what happens in very many trials, and we must actually observe many trials to pin down probability. Probability gives us a language to describe the long-term regularity of random behavior. The Practice of Statistics, 5th Edition 5 The Idea of Probability The law of large numbers says that if we observe more and more repetitions of any chance process, the proportion of times that a specific outcome occurs approaches a single value. The probability of any outcome of a chance process is a number between 0 and 1 that describes the proportion of times the outcome would occur in a very long series of repetitions. The Practice of Statistics, 5th Edition 6 Check your understanding 1. According to the Book of Odds Web site, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61. a) Explain what the probability 0.61 means in this setting. b) Why doesn’t this probability say that if 100 U.S. adults are chosen at random, exactly 61 of them usually eat breakfast? 2. Probability is a measure of how likely an outcome is to occur. Match one of the probabilities that follow with each statement. 0 0.01 0.3 0.6 0.99 1 a) The outcome is impossible. It can never occur. b) The outcome is certain. It will occur on every trial. c) This outcome is very unlikely, but it will occur once in a while in a long sequence of trials. d) This outcome will occur more often than not. The Practice of Statistics, 5th Edition 7 Myths About Randomness The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. 1. Toss a coin six times and record heads (H) or tails (T) on each toss. Which of the following outcomes is more probable? HTHTTH TTTHHH – Both are equally likely. – Heads and tails are equally probable says only that about half of a very long sequence of tosses will be heads. It doesn’t say that heads and tails must come close to alternating in the short run. – The coin has no memory. It doesn’t know what past outcomes were, and it can’t try to create a balanced sequence. The Practice of Statistics, 5th Edition 8 Myths About Randomness The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. 2. A couple has had 7 children, all girls, they decided to have another child because they must be due for a boy. Are they more likely to have a boy than a girl? The Practice of Statistics, 5th Edition 9 Myths About Randomness The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. The myth of short-run regularity: The idea of probability is that randomness is predictable in the long run. Our intuition tries to tell us random phenomena should also be predictable in the short run. However, probability does not allow us to make short-run predictions. The myth of the “law of averages”: Probability tells us random behavior evens out in the long run. Future outcomes are not affected by past behavior. That is, past outcomes do not influence the likelihood of individual outcomes occurring in the future. The Practice of Statistics, 5th Edition 10 Simulation The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. Performing a Simulation State: Ask a question of interest about some chance process. Plan: Describe how to use a chance device to imitate one repetition of the process. Tell what you will record at the end of each repetition. Do: Perform many repetitions of the simulation. Conclude: Use the results of your simulation to answer the question of interest. We can use physical devices (e.g. slips of paper, die), random numbers (e.g. Table D), and technology to perform simulations. The Practice of Statistics, 5th Edition 11 Example: Simulations with technology In an attempt to increase sales, a breakfast cereal company decides to offer a NASCAR promotion. Each box of cereal will contain a collectible card featuring one of these NASCAR drivers: Jeff Gordon, Dale Earnhardt, Jr., Tony Stewart, Danica Patrick, or Jimmie Johnson. The company says that each of the 5 cards is equally likely to appear in any box of cereal. A NASCAR fan decides to keep buying boxes of the cereal until she has all 5 drivers’ cards. She is surprised when it takes her 23 boxes to get the full set of cards. Should she be surprised? Problem: What is the probability that it will take 23 or more boxes to get a full set of 5 NASCAR collectible cards? The Practice of Statistics, 5th Edition 12 Example: Simulations with technology Plan: We need five numbers to represent the five possible cards. Let’s let 1 = Jeff Gordon, 2 = Dale Earnhardt, Jr., 3 = Tony Stewart, 4 = Danica Patrick, and 5 = Jimmie Johnson. We’ll use randInt(1,5) to simulate buying one box of cereal and looking at which card is inside. Because we want a full set of cards, we’ll keep pressing Enter until we get all five of the labels from 1 to 5. We’ll record the number of boxes that we had to open. The Practice of Statistics, 5th Edition 13 Example: Simulations with technology 3 5 2 1 5 2 3 5 4 9 boxes 5 1 2 5 1 4 1 4 1 2 2 2 4 4 5 3 16 boxes 5 5 5 2 4 1 2 1 5 3 10 boxes 4 3 5 3 5 1 1 1 5 3 1 5 4 5 2 15 boxes 3 3 2 2 1 2 4 3 3 4 2 2 3 3 3 2 3 3 4 2 2 5 22 boxes Conclude: We never had to buy more than 22 boxes to get the full set of NASCAR drivers’ cards in 50 repetitions of our simulation. So our estimate of the probability that it takes 23 or more boxes to get a full set is roughly 0. The NASCAR fan should be surprised about how many boxes she had to buy. The Practice of Statistics, 5th Edition 14 Check your understanding In the NASCAR and breakfast cereal example, what if the cereal company decided to make it harder to get some drivers’ cards than others? For instance, suppose the chance that each card appears in a box of cereal is Jeff Gordon, 10%; Dale Earnhardt, Jr., 30%; Tony Stewart, 20%; Danica Patrick, 25%; and Jimmie Johnson, 15%. How would you modify the simulation in the example to estimate the chance that a fan would have to buy 23 or more boxes to get the full set? The Practice of Statistics, 5th Edition 15