ࡱ>      bjbjcTcT 4J>>+===QQQ8,Qs`q q q q q """jslslslslslslsQvxls=""@"""lsq q 7s4n'n'n'"q =q jsn'"jsn'n'bn$T6sq @9mQ&vqVVss<sr.yR'y6sy=6s ""n'"""""lslsn'"""s""""y""""""""" : Sum of two diceIntroduction: Students are give a situation in which two dice are rolled 36 times, and the numbers added, students are asked to predict the shape of the theoretical probability distribution for the set of outcomes. Problem/Plan Philip predicts a uniform distribution with each outcome of 2 to 12 appearing 3 or 4 times. Note: Jenny does not do a prediction on the shape of the distribution. Jenny uses an appropriate recording model to find all the possible outcomes and suggests the distribution would be uneven. Plan should be to model theoretical distribution and test how many trials it takes to get the experimental distribution close to to it. (Students could predict how many trials are needed.) Plan should be to use software to test this. Students state assumptions: the dice are fair, the dice are independent of each other, the rolling is fair 91 Student carries out trials of the experiment, accepting that small samples often result in considerable variation from the predicted result. Student suggests combining class results to form a larger sample and accepts that as the sample size becomes bigger, the experimentally estimated result resembles the theoretical result better.Students should predict an experimental distribution as well.  Analysis     Use software to show that when the number of trials is small there is great variabilty in the experimental distributions but as the number of trials gets very large the experimental distribution settles down and is very close to the theoretical distribution. Discuss converse: If dont know or cannot work out the theoretical distribution what should you do to estimate the theoretical probabilities? Conclusion  Students accept the concept of the natural variation in individual samples. Students can express experimentally estimated and theoretical probabilities as fractions, decimals and percentages.      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