ࡱ> b kjbjb ~+$    @ @ @  8#8#8#8p##l {$$$$$I'I'I',{.{.{.{.{.{.{,U}RZ{@ I'&|I'I'I'Z{ +  $$o{ + + +I' $@ $,{ +T $x     I',{ + +~t8 T@ y# 8#E)4xXyL{0{xTY +Yy +@ D d Fair DiceIntroduction: Key idea: Even with a fair die and a large number of trials, random variation occurs. Trials can be used to develop criteria for fairness of a die. Problem The class is investigating how they can tell if a die is fair.Plan Grace draws a bar graph of the theoretical distribution of the roll of one die, and checks that her graph agrees with those of other students. They agree that the theoretical distribution of the roll of one die is rectangular, each bar is the same height. They realize that they will need to take a sample of dice rolls in order to compare the experimental distribution with the theoretical distribution. They discuss how large a sample they will need and start with a sample of 12 rolls. Data/ Analysis  Grace and Shayal roll a fair dice 12 times, making a dot plot of the results. They repeat the process 3 times and compare their results with those of other students, noticing what is the same and what is different, and predicting what the distribution of an unfair die would look like. They notice that there is sometimes zero frequency for a face number in 12 rolls of a fair die, but no number turns up more than 4 times, so they predict that a die with one number showing up 5 or more times in 12 rolls would be unfair. They accept the concept that there is natural variation in individual samples. They carry out simulations to repeatedly roll a virtual die 12 times, displaying a frequency distribution of the results on the board. There are computer simulations available fairdice.xls and fairdice.ftm. They discuss what they notice about the results. They notice that in 50 simulations of 12 dice rolls, there are occasionally 5 rolls of the same number, but never 6, so they adjust their prediction to predict that if the same number turns up half or more of the time in 12 rolls, then the die would be unfair. Problem The teacher asks them to predict what the distribution of 30 rolls of a die would look like. They predict that samples of 30 will have no numbers with zero frequency or turn up more than 10 times. Data/ Analysis  They roll a die 30 times and make a dot plot, comparing their result with other students graphs, and they observe that the result is in line with their predictions. By using spreadsheet, graphics calculator, scientific calculator or other random function, students carry out 50 simulations of 30 dice rolls and notice that the same number does occasionally occur more than 40% of the time and that there are occasionally numbers that have zero frequency in 30 rolls. They decide to modify their prediction to predict that if a number that turned up 50% of the time in 30 rolls, the die would be unfair. Problem/Plan The class discusses the influence of sample size on distribution to answer the question, How many times should you roll a die to decide if you think it is fair or not? They decide that they would have to roll a die at least 120 times in order to be able to decide if a die was really fair. Data/ Analysis  They observe the demonstration of 50 simulations of 100 die rolls. Grace and Shayal notice that the result for any one number is never less than 5% or greater than 30%. They decide that they would consider a die to be unfair if any number turns up less than 5 times in 100 rolls, or if a number turned up in more than 30 out of 100 rolls. Problem/Plan Grace and Shayal are given a soft fuzzy die and are asked to decide if it is fair and to justify their decision. They roll the die 100 times, and make a graph of the distribution of rolls.Data/ Analysis They notice that 2 turns up only 5 times in 100 rolls and 6 turns up 30 times but report that the result could have happened by chance for a fair die, so they do not conclude that the fuzzy die is unfair. They do not have sufficient evidence to support the die being unfair.Conclusion The class discusses the fact that a biased die might roll the same number every time, but a slightly biased die would give a distribution just slightly different from an unbiased die. They acknowledge that the sample size you would take to test a die would depend on how biased the die was, and how important it was to you to know that a die was fair.ReflectionExtension activityIf a die was biased, it may be because one face has an extra weight on it. Explain how to test if the other numbers occur equally often. What would you expect and why? Research the illegal Australian game called two-up. Investigate outcomes if a biased coin was used. Explain the game and justify your conclusions.      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