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lOURNAL OF SCIENCE OF HNUE Interdisciplinary Science, 2014, Vol 59, No 5, pp 64 70 This paper is available online at http //stdb hnue edu vn USING MONTE CARLO SIMULATION WHEN TEACHING PROBABILITY TO H[.]
lOURNAL OF SCIENCE OF HNUE Interdisciplinary Science, 2014, Vol 59, No 5, pp 64-70 This paper is available online at http://stdb.hnue.edu.vn USING MONTE CARLO SIMULATION WHEN TEACHING PROBABILITY TO HIGH SCHOOL STUDENTS Nguyen Phuong Chi Department of Mathenuitics Education, Hanoi National University of Education Abstract Probabdity is an unportant topic in die high school madiematics curriculum However, dealing with randomness is always a challenge for Vietnamese students because they have not been encoiu.aged to use dieir intuitive abilities and they lack the experience needed to feel the Hkelihood of probabilities To overcome this difficulty, it is necessary to use simulation in the teaching and learning probability at school This paper would like to introduce Monte Carlo simulation which is a method of solving probabiUty problems through the use of experiments This mediod should be used in teaching probabiUty to high school students because it increases theu practical experience and it teaches students how to apply probability when attempting to solve real world problems This paper explains what Monte Carlo simulation is and why and how teachers should teach Monte Carlo simulation to high school students Keywords: Monte Carlo simulation, model, experiment, teaching probability, high school students Introduction Probability is an appropriate topic in die school madiematics curriculum and textbooks because it is an mdispensable part of real Ufe, it suppUes necessary tools to comprehend die world around us and it provides meaningful appUcations of mattiematics at aU levels [1, 6, 9, 10, 11 and 13] In addition, dus topic can contiibute to die mental development of stiidents [5, andll] and it is inherenfly interesting, exciting, and motivating for most students [13] Aldiough probabiUty plays an important role in die teaching of madiematics at school, it is always a difficuU for stiidents due to die cognitive demands of deaUng widl randomness in conttast widi die deterministic dunking associated wifli most uses of Received November 05,2013 Accepted June 19,2014 Conact Nguyen Phuong Chi, e-mail address: chinp@hiiue.edu.vn 64 Use Monte Carlo simulation in teaching probability for High school students mathematics [8] Coping with variabiUty, samples, random trials, centers and distribution is really a challenge for high school smdents because their abiUty to intait is low and they have not previously been shown that most of the real world is in the realm of calculable probabiUty One way to help students improve their abiUty to intuit and use probabiUty is to use simulation to provide them with an opportanity to obtain practical experience Simulation helps students understand how probabiUty is applied and how to grasp real life situations Using simulation to teach probabiUty at school is supported by educators around the world Shaughnessy, an American educator, recommended that the smdy of probability in schools should rely on simulation to model experiments that require the use of problem-solving techniques [12] A Swiss educator named Inhelder said that simulation helps students discover and develop probabiUstic truths in realistic problems [7] Wolpers and Gotz, two German educators, stated that students must first experience probability situations to be able to understand models and such experience can be best achieved through the use of simulations [16] Another American educator named Bryan said that practical problems from the simplest to the most complex can be solved, or at least approximate answers can be found, using simulation He also said that simulation is an ideal mechanism for providingtiieteacher with the opportanity to develop a systematic progression from estimating probabilities to drawing conclusions and making inferences [2] A look at Vietaamese high school textbooks shows taat there is no use of simulation to create probabilistic sitaations ([3]) This absence of sunulation in the textbook and tiie teaching of probabiUty deny stadents an opportunity to gain practical experience and apply probabiUstic knowledge in the real world This is not in Une with tiiat which is stated in the Education Law of Vietnam: "Educational content has to relate to real life" and 'Teaching methods have to teach students the how to apply inrealUfe the information they have leamed in school" [4] The authors of diis study hope that teachers will begin to use Monte Carlo simulation in order to improve the teaching and learning of probabiUty at the high school level The efficiency of this method has been shown by educators around the world ([2], [7], [12], [14], [15], [16]) 2.1 Content What is Monte Carlo simulation? Consider tfie foUowing problems in probabiUty (these problems are based on problems presented in Waticins [15] and Travers [14]): Problem 1: Each box of milk contains one of seven kinds of toys A stadent wants to know how many boxes of milk he should expect to buy to get the entire set of toys 65 Nguyen Phuong Chi Sincettiisis a difficuU problem to solve analytically, die student decides to solve it using simulation The smdent puts die names of die seven toys on slips of paper and put die slips into a box He wididraws one slip of paper, writes down die name of die toy and replaces die slip of paper This is continued until die name of each toy is drawn The number of boxes 'purchased' (sUps drawn) is recorded Repeating the process twentytimes,it is found diat die average number of box diat must be purchased is 18.7 Problem 2: Minhfliinkshe doesn't have to stiidy prior to taking tests He is willing to take his chances Suppose diat he take atine-falsetest in Geography and doesn't know die answers totenof die questions What's die chancefliathe'll get seven or more of fliose ten questions correct by guessing? To solve this problem, he uses a coin with heads meaning "Minh has the correct answer" and tails meaning "Minh has the wrong answer." Minh tosses the coin ten times and count, the number oftimesit came up heads If it came up heads seven of more times, he writes down "Minh has seven or more correct answers." When this process is repeated 100times,it is found that Minh has seven or more correct answers 21 of the 100 times 21 Thus P (seven or more conect answers) is, for these 100 sets of tossed, —— or 0.21 or 21% Problems and above have been solved by a technique called Monte Carlo simulation This is a method of solving probabilistic problems experimentaUy It involves finding a model for the given problem The model is physically different, it is easier to operate, and it has the same mathematical characteristics as the original problem The distinctive feature of Monte Carlo simulation is the use of objects such as dice or coins The dieoretical basis for the Monte Carlo method-is caUed the law of large numbers, which states that the moretimesa simulation is run (number of successes) / (number of runs) The closer one gets toflieactual analytical probabiUty 2.2 Why teach Monte Carlo simulation? Monte Carlo simulation should be taught for the foUowing reasons: - First, as a type of simulation or madiematical model, dus method teaches stiidents how torepresentreal-worldsystems in terms of mathematical relationship One of die most unportant aims ofteachingprobabiUty is to help stiidents leam how to solve various problems m real Ufe and Monte Carlo simulation can it efficiently Many real Ufe problems can be solved using Monte Carlo simulation More specifically, almost aU of any probabiUty or expected-value problems can be solved using an appropriate Monte Carlo sunulation [15]: + One basic type of problem involves determining die probabiUty of success or failure For example, m problem presented above, success to Minh is when he guesses 66 Use Monte Carlo simulation in teadiing probability for High school students seven or more correct answers + A second basic type of problem asks for an ^pected value, not a probabiUty For example, in problem presented above, smdents must answer the question: "How many milk boxes he can ^pect to have to buy in order to get the entire set of toys?" Therefore, knowing how to use Monte Carlo simulation can help smdents solve a large class of real Ufe problems This method provides stadents with an opportanity to use probabiUty to understand certain aspects of real Ufe sitaations - Second, Monte Carlo simulation is a useftil way to verily the results obtained from a purely analytic explanation For example, if the probabiUtyfliata baby wiU be a boy is ^ and the probabiUty tiiat the baby wiU be a girl is - , isflieprobabiUty of having two boys in a family of two children - or - ? That is, is tiie sample space BB, BG or GG, or is it BB, BG, GB GG? (B is boy, G is girl) Stadents can be convinced that it is the latter byflippingtwo coins and suppose that heads means 'baby boy', tails means 'baby girl' If this is done 100 times, a distribution such as HH 23 HT 48 TT 29 will appear, indicating that these three outcomes are not equaUy likely [15] - Third, Monte Carlo is easy to in the classroom because most of the materials needed are at hand Moreover, stadents find this kind of mathematics to be frm and interesting and t h ^ enjoy trying to device new and clever variations [15] 2.3 How should Monte Carlo simulation be taught to high school students? J.l Students should be shown and given the chance to practice the general steps of Monte Carlo sunulation To use Monte Carlo simulation, teachers should teach smdents how to represent a real world problem in terms of probabiUty and then solve that problem The best way is to show stadents the general steps of Monte Carlo simulation so that t h ^ can foUow these steps when they face real Ufe problem that are solvable using this method Travers [14] Usts five general steps for the Monte Carlo approach to probabiUty They are: - Model: Find an appropriate model for the problem sitaation For example, in problem flie situation was modeled by using a coin, where heads means "Minh has a correct answer" and tails means '*Minh has a wrong answer" 67 Nguyen Phuong Chi - Trial: Determine what constitiites a tiial consist of Oftentimes a ttial consists of tossing die com (orroUingdie die) until a predetermined number of outcomes is obtained For example, m problem 2, a trial consists of tossing die coin ten times, onceforeach test question - Successful trial: Determine whether or not a trial is successfid For example, in problem 2, a successfid tiial occurs when the heads appears at least seventimes,which means seven or more correct answers are obtained - Number of trials: The tiials arerepeateduntil die predetermined number of ttials has been completed For most problems undertaken in school, one hundred trials will provide adequate accuracy In problem 2, for example, one hundred trials should be performed - Probability (success): Estimate the probabiUty of a successful trial P (success) by theratio(number of successfid trials) / (total number of trials) For example, in problem 21 2, the probabiUty of getting seven or more correct answers is estimated by the ratio — These five steps need to be practiced as a problem solving techiuque by stiidents through various exercises The demonstrative exercise is presented below: Problem 3: What is the probabiUty that in a group offourpeople chosen at random, two or more were bom in the same month? Solutioii: Model: Use a twelve-sided die (one side for each month of the year) Trial: A trial consists of rolUng the diefourtimes,one for each person in the group Successful trial: A successfiil trial is one in which a number obtained more dian once in the four roUs offliedie -fliatis, at least two people have the same bkth mondi Number of trial: Repeat the tiial at least 100 times Probability (success): The probability P that at least two people share die same birth month is estimated by the ratio: number of success/number of trials 2.3.2 Use Monte Carlo simulation with the support of computer software Simulations can be rapidly produced with help of computer programs Without computer sunulation, gadiering sufficient experimental data to investigate problems would be sotime-consumingdiat it would not be feasible for the classroom [7] At die high school level, Monte Carlo simulation can be performed more effectively widl die support of Excel software Excel software is quite famiUar to high school smdents and fliis can help fliem simulate tossing a coin or rolling a die Widi this software, die stiidents can toss a coin or rod a die diousandstimesrepresentationaUyrather dian actually doing it with their hands For instance, to solve problem above, instead of using a real twelve-sided die we can model rolUngflusdie wiflifliesupport of EXCEL software as foUow: 68 Use Monte Carlo simulation in teaching probability for High school students Simulation of roUing a twelve-sided die In our case, a die has twelve faces Hence, we use function RANDBETWEEN (1,12) This function wiU return a random integer between and 12 To simulate 1000roUsof die die, we type Al= RANDBETWEEN (1, 12), for A2 to AlOOO we only need to use F4 or Cttl-D to copy the formula of A1 Then we have column A whichrepresents1000 roUs of the twelve-sided die Conclusions Using simulation in teaching and learning probabiUty at school is a trend worldwide As a type of simidation, Monte Carlo simulation teaches students to model and solve many real world problems This method can help smdents by giving them practical experience, improving their abiUty to experiment and helping them understand how probabiUty is appUcable in real Ufe High school smdents should leam about and practice the general steps of Monte Carlo simulation so that they can use this method efficienfly whenever they face a probabiUstic problem In addition, students should make use of computer software such as Excel to perform the simulation more conveiuendy REFERENCES [1] Ben-Zvi, D., Garfield, J., 2004 Statistical Literacy, Reasoning, and Thinking: Goals, Definitions, and ChaUenges In D Ben-Zvi, J Garfield (Ed.), The ChaUenge of Developing Statistical Literacy, Reasoning and Thinking (pp.3-15) Dordrecht: Kluwer [2] Bryan, B.,1989 Using Simulation to Model Real-World Problems In R Morris (Ed.), Studies in Mathematics Education: The Teaching of Statistics, Volume (pp.94-100) Paris: Unesco [3] Doan Quynh, Nguyen Huy Doan, Nguyen Xuan Liem, Nguyen Khac Minh, Dang Hung Thang, 2009 Dai so va giai tich 11 nang cao Nxb Giao Due [4] Education Law, 2005 Trong Tim hieu Luat Giao due 2005 (pp.24-70)] Ha n5i: Nha Xuat Ban Giao Due [5] Fischbein, E., 1975 The Inmitive Sources of ProbabiUstic Thinking in Children Dordrecht Boston: D Reidel PubUshing Company [6] Freudendial, H., 1970 The Anns of Teaching ProbabiUty In L Rade (Ed.), The Teaching of Probability and Statistics (pp.I51-168) Stockholm: Almqvist and WikseU [7] Inhelder, W, 1981 Solving ProbabiUty Problems through Computer Simulation In A.P Shulte, J.R Smart (Eds.), Teaching Stiitistics and ProbabiUty (pp.220-224) America: The National Cotmcil of Teachers of Mathematics 69 Nguyen Phuong Chi [8] Jones, G A., 2005 Explormg ProbabiUty m School: ChaUenges for Teaching and Learning New York: Springer [9] Kapadia, R., Borovcndc, M., 1991 The Educational Perspective In R Kapadia, M Borovcnik (Eds.), Chance Encounters: Probability in Education (pp.1-26) Dordrecht, Netherlands: Kluwer Academic PubUshers [10] Pereua-Mendoza, L., Swift, J., I98I Why Teach Statistics and Probabdity - A Rationale In A P Shulte, J R Smart (Eds.), Teaching Statistics and Probability (pp.1-7) America: The National Councd of Teachers of Madiematics [11] Renyi, A., 1987 A Diary on Information Theory Chichester New York Brisbane Toronto Singapore: John Wiley, Sons [12] Shaughnessy, J M., 1981 Misconceptions of Probability: From Systematic Enorsto Systematic Experiments and Decisions In A.P Shulte, J.R Smart (Eds.), Teaching Statistics and ProbabiUty (pp.90-100) America: The National CouncU of Teachers of Mathematics [13] Shulte, A P., Smart, J R., 1981 Teaching Statistics and ProbabiUty America: The National Councd of Teachers of Mathematics [14] Travers, K J., I98I Using Monte Carlo Metiiods to Teach Probability and Statistics In A.P Shulte, J.R Smart (Eds.), Teaching Sutistics and ProbabiUty (pp.210-219) America: The National Council of Teachers of Mathematics [15] Waddns, A E., 1981 Monte Carlo Simulation: ProbabiUty die Easy Way In A.R Shulte, J.R Smart (Eds.), Teaching Statistics and ProbabiUty (pp.203-209) America: The National Council of Teachers of Mathematics [16] Wolpers, H., Gotz, S., 2002 Madiematikunterricht in der Sekundarstiife n Band 3: Didaktik der Stochastik Braunschweig / Wiesbaden: Vieweg 70 ... should Monte Carlo simulation be taught to high school students? J.l Students should be shown and given the chance to practice the general steps of Monte Carlo sunulation To use Monte Carlo simulation,... original problem The distinctive feature of Monte Carlo simulation is the use of objects such as dice or coins The dieoretical basis for the Monte Carlo method-is caUed the law of large numbers,... (number of runs) The closer one gets toflieactual analytical probabiUty 2.2 Why teach Monte Carlo simulation? Monte Carlo simulation should be taught for the foUowing reasons: - First, as a type of