Page 34 - Demo

xxx Randomness, Probability, and%u00a0Simulation How many boxes of cereal to collect them all? Using simulations to estimate probabilities PROBLEM: In an attempt to increase sales, a breakfast cereal company decides to offer a promotion. Each box of cereal will contain a collectible comic book featuring a popular superhero: The Flash, Batman, Superman, Wonder Woman, or Green Lantern. The company claims that each of the 5 comic books is equally likely to appear in any box of cereal. A superfan buys boxes of the cereal until they have all 5 superhero comic books. The fan is surprised when it takes 23 boxes to get the full set of comic books. Does this outcome provide convincing evidence against the company%u2019s claim that the 5 comic books are each equally likely to appear in any box of cereal? To help answer this question, we perform a simulation to estimate the probability that it will take 23 or more boxes to get a full set of the superhero comic books, assuming that the company%u2019s claim is true. (a) Describe how to use a random number generator to perform one trial of the simulation. We carried out 100 trials of the simulation and noted the results. The dotplot shows the number of cereal boxes it took to get all 5 superhero comic books in each trial. Simulated number of boxes required0 5 10 15 20 25 30 35 40 (b) Explain what the dot at 18 represents. (c) Use the results of the simulation to estimate the probability that it will take 23 or more boxes to get a full%u00a0set of comic books. (d) Based on the actual result of 23 boxes and your answer to part (c), is there convincing evidence that the 5%u00a0superhero comic books are not equally likely to be found in each box? Explain your reasoning. SOLUTION: (a) Let 1= The Flash; 2= Batman; 3= Superman; 4= Wonder Woman; 5= Green Lantern. Generate a random integer from 1 to 5 to simulate buying one box of cereal and determining which comic book is inside. Keep generating random integers until all 5 labels from 1 to 5 appear. Record the number of boxes it takes to get all 5%u00a0superhero comic books. (b) One trial where it took 18 boxes to get all 5 superhero comic books. (c) Because 3 of the 100 dots are 23 or greater, the probability %u2248 = 3/100 0.03 = . (d) Because it is unlikely (probability %u2248 0.03 ) that it would take 23%u00a0or more boxes to get a full set by chance alone when the superhero comic books are equally likely to be included in each box, this result provides convincing evidence that the 5 superhero comic books are not equally likely to be found in each box of cereal.EXAMPLE 2. Perform many trials. The simulation gives an estimate of the probability, so be sure to use an approximately equal sign. 1. Describe how to set up and use a chance process to perform one trial of the simulation.Identify what you will record at the end of each trial. 3. Use the results of your simulation to answer the question of interest.FOR PRACTICE TRY EXERCISE 13. Worked examples help you understand concepts and techniques cereal company decides to offer a promotion. Each box of cereal will contain a collectible comic book featuring a popular superhero: The Flash, Batman, Superman, Wonder Woman, or Green Lantern. The company claims that each of the 5 comic books is equally likely to appear in any box of cereal. when it takes 23 boxes to get the full set of comic books. Does this outcome provide convincing evidence against the company%u2019s claim that the 5 comic books are each equally likely to appear in any box of cereal? To help answer this question, we perform a simulation to estimate the probability that it will take 23 or more boxes to get a full set of the superhero comic books, assuming that the company%u2019s claim is true. (a) Describe how to use a random number generator to perform one trial of the simulation. EXAMPLES . Each example is closely tied to one of the Learning Targets listed at the beginning of the lesson. (b) Explain what the dot at 18 represents. (c) Use the results of the simulation to estimate the probability that it will take 23 or more boxes to get a full%u00a0set of comic books. (d) Based on the actual result of 23 boxes and your answer to part (c), is there convincing evidence that the 5%u00a0superhero comic books are not equally likely to be found in each box? Explain your reasoning. SOLUTION: MODEL SOLUTIONS. Each example features a solution that looks like handwriting to show you how to write a complete, accurate solution. Use this as a model to answer similar exercises. 45 (c) Use the results of the simulation to estimate the probability that it will take 23 or more boxes to get a (d) Based on the actual result of 23 boxes and your answer to part (c), is there convincing evidence that the 5%u00a0superhero comic books are not equally likely to be found in each box? Explain your reasoning. THE VOICE OF THE TEACHER. Study the worked examples and pay special attention to the %u201cTeacher Talk%u201d boxes that guide you step by step through the solution. These comments offer lots of good advice %u2014 as if your teacher is working directly with you to solve a problem. PRACTICE! Take time to work the paired problem in the Lesson Exercise set. Practicing a similar problem after studying the worked example helps you know whether or not you%u2019ve %u201cgot it.%u201d %u00a9 Bedford, Freeman & Worth Publishers. For review purposes only. Do not distribute.