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Status for AP Congrats! • We are done with Part II of the Topic Outline for AP Statistics! (10%-15%) of the AP Test can be expected to cover topics from chapter 4. • Coming up… Part III (20%-30%) Anticipating Patterns: Exploring random phenomena using probability and simulation Randomness, Probability, and Simulation Section 5.1 Reference Text: The Practice of Statistics, Fourth Edition. Starnes, Yates, Moore Consider This… • The “1 in 6 wins” Game • As a special promotion for its 20-ounce bottles of soda, a soft drink company 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.” 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. Is this group of friends just lucky, or is the company’s 1-in-6 claim inaccurate? Consider This… • For now, lets assume the company is telling the truth, and that every 20-ounce bottle of soda it fills has a 1-in-6 chance of getting a cap that says, “you’re a winner” • We can model the status of an individual bottle with a six-sided die: • Let 1 to 5 represent “please try again!” • 6 represent “you’re a winner” Activity! 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. Repeat step 1 four more times. In your five repetitions of this simulation, how many times did three or more of the group win a prize? 3. Combine results with your classmates. What percent of the time did friends come away with three or more prizes, just by chance? 4. Based on your answer in step 3, does it seem plausible that the company is telling the truth, but that the seven friends just got lucky? Objectives 1. The Idea of Probability -Law of Large Numbers -Probability 2. Myths about Randomness – Runs “hot hand” – “law of averages” 3. Simulations – Preforming simulations • State • Plan • Do • Conclude Idea of Probability The big fact emerges that when we look at random sampling or random assignment closely: chance behavior is unpredictable in the short run, but has a regular and predictable pattern in the long run. This remarkable fact is the basis of probability! Law of Large Numbers • The law of large numbers says that if we observe more and More and More repetitions of any chance process, the proportion of times that a specific outcome occurs, will approach a single value. • The fact that the proportion of heads in many tosses eventually closes in on .5 is guaranteed by the law of large numbers. – With more and more times the situation is being conducted, the pattern emerges into what's known as the probability. Flipping a Coin: Probability of Heads? Probability • The probability of an outcome of a chance process is a number between 0 and 1 that describes the proportion of times that outcome would occur in a very long series of repetitions. 0….……………… (.5)………………….1 0% ………………(50%)……………..100% Facts: Outcomes that never occur have a probability of 0 An outcome that happens on every repetition has probability of 1 Check for Understanding • According to the books of odds, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61 – Explain what probability 0.61 means in this setting – Why doesn’t the probability say that if 100 U.S adults are chosen at random, exactly 61 of them usually eat breakfast? Check for Understanding • Probability is a measure of how likely an outcome is to occur. Match one of the probabilities that follow with each statement. Be prepared to defend your answer 0 0.01 0.3 0.6 0.99 1 a) This outcome is impossible. It can never occur. b) This 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. Myths About Randomness The idea of probability seems straightforward. It answers the question: “what would happen if we did this many times?” • Both the behavior of random phenomena and the idea of probability are a bit subtle. – We meet chance behavior constantly in our daily lives… and psychologists tell us that we deal with it poorly… Myths About Randomness • Unfortunately, it is our intuition about randomness that tries to mislead us into predicting the outcome in the short run… • 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! • Almost everyone says that HTHTTH is more probable because TTTHHH does not “look random.” • In fact both are equally likely. The coin has no memory, it doesn’t know what past outcomes were, and it cant try to create a balanced sequence. • Myths about Randomness 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. Randomness, Probability, and Simulation The idea of probability seems straightforward. However, there are several myths of chance behavior we must address. Myth: “Runs” • The outcome TTTHHH in tossing six coins looks unusual because of the runs of 3 straight tails and 3 straight heads. • Runs seem “not random” to our intuition but are quite common. • More examples: – “she’s on fire!” (basketball player makes several shots) – “Law of Averages” Roulette tables in casinos. “red is due!” from the display board Simulations and Performing Them • Simulation: The imitation of chance behavior, based on a model that accurately reflects the situation is called a simulation Simulations and Preforming Them Statistics Problems Demand Consistency! • State: What is the question of interest about some chance process? • Plan: Describe how to use a chance device to imitate one repetition of the process. Explain clearly how to identify the outcomes of the chance process and what variables to measure. • 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, random numbers (e.g. Table D), and technology to perform simulations. Example: 6 in 1 Game! State Plan Do Conclude What’s the probability that three of seven people who buy a 20ounce bottle of soda win a prize if each bottle has a 1/6 chance of saying, “you’re a winner!”? Use a six-sided die to determine the outcome for each person’s bottle of soda. 6 = wins a prize 1 to 5 = no prize Roll the dice seven times, once for each person Record whether 3 or more people win a prize (yes or no) Each student perform 5 repetitions Out of 125 total repetitions of the simulation, there were 15 times when three or more of the seven people won a prize. So our estimate of the probability is 15/125, or about 12% Warning: Phrasing Your Conclusions • When students make conclusions, they often lose credit for suggesting that a claim is definitely true or that the evidence proves that a claim is incorrect. • A better response would be to say that there is sufficient evidence (or there isn’t sufficient evidence) to support a particular claim. We will cover this point is a much larger depth when we get to inference in the 2nd semester. Objectives 1. The Idea of Probability -Law of Large Numbers -Probability 2. Myths about Randomness – Runs “hot hand” – “law of averages” 3. Simulations – Preforming simulations • State • Plan • Do • Conclude Homework 5.1 Homework Worksheet Start Chapter 5 Reading Guide