ࡱ> /1. !bjbjcTcT 4J>>Y''';;;;$_;#```~#######{&)#'`>"```#E#`8'~#`~#r"TF#~eJ;X#j##0##2)@L)F#)'F#$```````##```#````)````````` :  Achievement Objective Investigate situations that involve elements of chance: applying distributions such as the Poisson, binomial, and normal. Exemplar 1 Arvind and Sheba have three different coloured squares representing three songs on an iPod. They list the number of ways they can listen to the three songs. They extend the idea to five songs and develop the idea of factorials, with n! for the number of ways of ordering n objects using the multiplication principle, They then consider the possible ways of listening to three of the five tracks, and hence develop a formula for calculating the number of permutations of r items selected from n realizing that the order of playing the tracks is important. They use the multiplication principle for independent events to calculate the number of ways. 5 x 4 x 3 = 60 To confirm the idea they list the number of ways a family which borrows 10 DVDs can watch four of them on one day. 10 x 9 x 8 x 7 Their teacher helps them to generalize the formula for nPr using factorials. They confirm by enumerating the outcomes for simple cases. Arvind and Sheba then work on calculating the number of ways a teacher can select three students out of five to go and collect books from the library. They realize that order does not matter in this situation, but that it is related to the number of permutations of three objects selected form 5. They note that there are six ways of ordering the group A,B and C, and six ways of ordering each of the groups of three, i.e. 3!. They compare this with the number of permutations and develop the formula for nCr leading to the formula for Combinations. These values can be related to the binomial coefficients in Pascals triangle, and confirmed using technology. Exemplar 2 David and Jonah each select five counters from a bag containing a large number of counters in two colours, say red and blue, where the proportion of red counters is known, replacing the counters after each withdrawal. They each count up the number of red counters in their selection of ten, and record the result. The player who has the higher number of red counters wins a point. If both players have the same number of red counters, neither scores a point. David and Jonah keep playing until one of them reaches 20 points. [This could be regarded as a simulation, for example, two traffic police checking for warrant of fitness, where a red counter represents no warrant of fitness, blue represents a current warrant.] Recording sheet # of red counters012345tallyllllllllllexperimental probability David and Jonah use their results to calculate the experimental probability for each of the possible outcomes and draw a column graph of their results. They observe the shape and note that it appears to be approximately symmetrical about some average value. After conducting the experiment David and Jonah estimate the proportion of red counters in the bag, and discuss how many times they would need to repeat the experiment in order to be confident that their estimate was correct. David and Jonah then calculate the theoretical probability for each outcome, using a probability tree and their estimate of the population proportion of red counters (or the actual proportion given by their teacher) or p. Alternatively, they could use the principles developed during explorations of combinations. They can check their values using their calculator. # of red counters012345theoretical probability(1-p)55p(1-p)410p2(1-p)310p3(1-p)25p4(1-p)p5 David and Jonah discuss the patterns they can see in the theoretical probabilities, leading to the general formula for k successes out of n trials Prob (X = k) = nCk pk (1-p)k Their teacher leads them to the realization that the coefficients 1, 5, 10, 10, 5, 1 are the binomial coefficients from Pascals triangle. David and Jonah compare the column graph for their experimental probabilities with a column graph of the theoretical probabilities. Discussing with the teacher, they could tease out the conditions for a binomial distribution to be a suitable model, that is Two possible outcomes, success or failure A fixed number of trials, n A fixed probability of success, p Each trial independent of the others Exemplar 3 Alice and Zoe play a game (which may be a simulation) tossing three dice, and counting how many sixes appear. A player wins 10 points for tossing three sixes, 5 points for tossing 2 sixes and 2 points for tossing one six. The first player to twenty points wins. They record their results carefully in a table. 3 sixes2 sixes1 sixzero sixestallyfrequency Alice and Zoe are aware that the number of sixes on their throw is a random variable, X, which can take a range of values and depends on chance. They discuss the conditions for their game with their teacher, who identifies the conditions for the Binomial distribution in the context of this game. Alice and Zoe then compare their empirical (experimental) results with the theoretical results from tables, graphs, calculator or computer. The general formula for P(X = x) can be confirmed using a probability tree Exemplar 4 David and Peter spend a lesson counting how many cars enter a petrol station during a five minute period, or how many people approach a money machine during a five minute time interval at a particular time of day, or by counting the number of chocolate chips in a cookie, or by counting the number of sharks in a fixed area of the ocean on an applet. They tally their results and plot a bar (column) graph of the results. They notice that the average number of cars arriving in a five minute interval is obvious from the graph, and recognize this value as a rate which would be proportional to the time interval. They calculate the mean and standard deviation of their results, perhaps noting that the standard deviation is approximately the square root of the mean. They compare their experimental probabilities with those calculated using technology or the formula  EMBED Equation.3 . They discuss how the situation they are considering now compares with the Binomial situation, notably that both involving counting, that is they are discrete distributions. They may recognise that there is an upper value to the values that x can take for binomial but not Poisson and that the second deals with a rate, the Binomial does not. The teacher summarises the conditions under which a Poisson distribution can be applied: Selections of 3 items from ABCDE ABC ACB BAC BCA CBA CAB 6 permutations of three items     ! % & . N i t p y  ǺǪtttttttttttttihjOnH*OJQJ^Jh8@hjOnOJQJ^Jh#)OJQJ^J hfhjOnOJQJ^JmH sH hjOnOJQJ^JmH sH h!hjOn5CJOJQJaJhphjOnOJQJ^J3h!hjOn5CJOJQJ^JaJmH nH sH tH hjOnOJQJ^J(hwhjOnCJOJQJ^JaJmH sH 'P , ?  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