Simulation AP STATISTICS “Statistics means never having to say you're certain.” Introduction Toss a coin three times. What is the likelihood of a run of three or more consecutive heads or tails? A couple plans to have children until they have a girl or until they have four children, whichever comes first. What are the chances that they will have a girl among their children? Three methods we can use to answer questions about chance Try to estimate the likelihood of a result of interest by actually observing the random phenomenon many times and calculating the relative frequency of the results. Develop a probability model and use it to calculate a theoretical answer. Start with a model that, in some fashion, reflects the truth about the random phenomenon, and then develop a plan for imitating (or simulating) a number of repetitions of the procedure. Simulating the birth of children A couple plans to have children until they have a girl or until they have four children, whichever comes first. What are the chances that they will have a girl among their children? Let a flip of a fair coin represent a birth, with heads corresponding to a girl and tails a boy. Assuming that girls and boys are equally likely to occur on any birth, the coin flip is an accurate imitation of the situation. If this coin flipping procedure is repeated many times, to represent the births in a large number of families, then the proportion of times that a head appears within the first four flips should be a good estimate of the true likelihood of the couple’s having a girl. Simulating the birth of children A single die could also be used to simulate the birth of a son or daughter. Let an even number represent a girl, and let an odd number represent a boy. Simulation The imitation of chance behavior, based on a model that accurately reflects the situation, is called a simulation. Performing a Simulation 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 variable 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. Simulation Steps 1. State the problem (model) or describe the random phenomenon and define the key components. Toss a coin 10 times. What is the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? (Optional on AP Exam) There are two: A head or tail is equally likely to occur on each toss. Tosses are independent of each other (that is, what happens on one toss will not influence the next toss.) Simulation Steps 2. Plan: Assign digits to represent outcomes. (State what you would record.) One digit simulates one toss of the coin. Odd digits represent heads; even digits represent tails. Successive digits in the table simulate independent tosses. 3. Do: Simulate many repetitions. (Conduct the trials. The AP Exam requires that you use a random number table.) Looking at 10 consecutive digits in a random number table simulates one repetition. Be sure to keep track of whether or not the event we want (a run of at least 3 heads or at least 3 tails) occurs on each repetition. Random Digit Table Digits: 19223 95034 05756 28713 96409 12531 H/T? HHTTH HHTHT THHHT TTHHH HTTTH HTHHH Runs of 3? Suppose that 22 additional repetitions were done for a total of 25 repetitions. 23 of them did have a run of 3 or more heads/tails. Simulation Steps 5. State your conclusions. (Usually only appropriate if you have at least 100 trials.) We estimate the probability of a run of size three by the proportion: 23 estimated _ probabilit y 0.92 25 Assigning digits Example: Choose a person at random from a group of which 70% are employed. One digit simulates one person: 0, 1, 2, 3, 4, 5, 6 = employed 7, 8, 9 = not employed Assigning Digits Example: Choose one person at random from a group of which 73% are employed. Now we must let two digits simulate one person: 00, 01, 02, …, 72 = employed 73, 74, 75, …, 99 = not employed ALSO ACCEPTABLE: 01, 02, 03, …, 73 = employed 74, 75, 76, …, 00 = not employed Assigning Digits EXAMPLE: Choose one person at random from a group of which 50% are employed, 20% are unemployed, and 30% are not in the labor force. There are now three possible outcomes. One digit simulates one person: 0, 1, 2, 3, 4 = employed 5, 6 = unemployed 7, 8, 9 = not in the labor force ALSO ACCEPTABLE: 0, 1 = unemployed 2, 3, 4 = not in the labor force 5, 6, 7, 8, 9 = employed Frozen Yogurt Sales Orders of frozen yogurt (based on sales) have the following relative frequencies: 38% chocolate, 42% vanilla, and 20% strawberry. We want to simulate customers entering the store and ordering yogurt. (Follow the 5 step process.) STEP 1: How would you simulate 10 frozen yogurt sales based on this recent history? Step 2: State the Assumptions Orders of frozen yogurt (based on sales) have the following relative frequencies: 38% chocolate, 42% vanilla, and 20% strawberry. We also assume that customers order one flavor only, and that the customer’s choices of flavors do not influence one another. Step 3: Assign Digits to Represent Outcomes We will do pairs of digits. Why? 00 to 37 = outcome chocolate (C) 38 to 79 = outcome vanilla (V) 80 to 99 = outcome strawberry (S) Step 4: Simulate Many Repetitions Suppose that we have the following sequence of random numbers: 19352 73089 84898 45785 This yields the following 2-digit numbers: 19 35 27 30 89 84 89 84 57 85 which correspond to the outcomes: C C C C S S S S V S Step 5: State your conclusions The problem only asked for the process, but let’s look at the results: We estimate that the probability of an order for chocolate to be 4/10=0.4, vanilla to be 1/10 = 0.1, and strawberry to be 5/10 = 0.5. However, 10 repetitions are not enough to be confident that our estimates are accurate.