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Probability Estimation Using Simulation

Probability Estimation Using Simulation
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2023-06-02 05:58
Probability & Statistics
This example demonstrates how to estimate probabilities of compound events through simulations. It illustrates the use of random numbers, spinners, and coin tosses to model frequencies for compound events, thereby estimating their probabilities. Key math concepts involved include probability & statistics and simulation methods. The construction utilizes GeoGebra commands and objects such as RandomUniform, Take, Zip, Sequence, Sum, CountIf, Corner, Polygon, onboardingTextOffset, Max, DotPlot, Unique, Length, Textfield, Numerator, Segment, VerticalText. Suitable for students in grades 6-8 to learn probability & statistics and develop data analysis skills.

GeoGebra Commands & Parameters

  1. RandomUniform(0, 0.99, 250) → allData, {0.7888599107408635, 0.8251243093627335, 0.8867114543886895, 0.16990109360285083, 0.29769783148559475, 0.20879695156832093, 0.27183522586178144, 0.5529823198029883, 0.20633825985783713, 0.2914189227796139, 0.48819568212932407, 0.3943738981129027, 0.13669219070322827, 0.6405342710310281, 0.7008858404271803, 0.43417269641503436, 0.7992201861404843, 0.5883839649274654, 0.4879438557372989, 0.6039026667482905, 0.9885803950794079, 0.4339710194605095, 0.8795499078408345, 0.14691605272013333, 0.8483808232039035, 0.588998861449881, 0.5194166031305749, 0.05441750354094489, 0.6511433884589705, 0.04249836471162619, 0.7013303037462291, 0.7387381521545902, 0.6291468997748456, 0.6478162179896313, 0.8288301628091099, 0.5585624002956937, 0.1998142565394015, 0.6804323300342966, 0.7444784097857093, 0.6648246870994223, 0.8857452488491127, 0.9558067374586754, 0.6670796828154915, 0.7309434481205026, 0.6943687801015467, 0.04312560495032519, 0.18054916647919247, 0.7369495927334289, 0.9654990370657116, 0.9534239756927033, 0.28690831486539053, 0.16389163733353623, 0.026828655616191042, 0.03910790665652582, 0.03810417158281231, 0.7826641747739767, 0.607576561593509, 0.48638052351125416, 0.2070019703057784, 0.42010743584018595, 0.9591321954858942, 0.6124966219646227, 0.6569930981135772, 0.7648638078542611, 0.7185975540270669, 0.6661285640902564, 0.5621895492382887, 0.8660541922397621, 0.06662717634801414, 0.5114198322801937, 0.23614254449433048, 0.5493498796071093, 0.6949012558774964, 0.6645363125312125, 0.6364280300208012, 0.4874676643961211, 0.81094046329644, 0.3656944031447909, 0.6352823918510716, 0.675503744580158, 0.9887813305482805, 0.779752984114809, 0.9027524443908017, 0.24584881562089325, 0.1632793071062828, 0.7145008646636848, 0.10853921610534241, 0.7926900901377087, 0.06349204603162427, 0.5777541743029893, 0.2418702213125659, 0.7823070833146119, 0.4167186103956794, 0.7422973881635551, 0.41532643896177013, 0.5925069874232514, 0.8693305350526641, 0.9615600388146495, 0.06857944534685924, 0.4226212843426341, 0.06352931797157817, 0.7587151091038844, 0.561460905578895, 0.3944993435630029, 0.6405997512989652, 0.4576183707451657, 0.6126912133723075, 0.6648853251107788, 0.031102249421884026, 0.8156771237146007, 0.16093559515159703, 0.770453817424458, 0.4307570095994625, 0.016344881918090595, 0.5886352150208651, 0.8386060207629943, 0.4375222267817661, 0.9724829299388658, 0.01627815956932454, 0.8398912291353383, 0.7605523672626067, 0.18131254202875938, 0.8454285657589674, 0.5016201250813069, 0.8953472210206177, 0.057100942572620085, 0.6329250037545152, 0.06687490185675397, 0.4914620221047904, 0.46908261469150647, 0.9561875416781018, 0.6686976025955967, 0.8096470382541078, 0.9094109638209035, 0.6205404774521398, 0.7157510176484893, 0.4442211835328345, 0.1591693103349094, 0.7238616327121786, 0.1631987534326123, 0.32385643907753353, 0.6396389957520948, 0.15171091420298208, 0.45093884321031363, 0.6597610095016999, 0.9311302451161972, 0.01737016080749189, 0.46065186098410416, 0.8344971455402392, 0.12272647502410815, 0.045943433947103915, 0.45338794736377797, 0.24735445664317063, 0.1532748839618571, 0.49982048883069047, 0.9606189212909679, 0.28332373979843406, 0.8668761172096716, 0.16439878029739477, 0.22381918177540341, 0.10227570277134533, 0.5049834445724947, 0.187987970851575, 0.5307166036681368, 0.4629477983250507, 0.21201663549659228, 0.058288190066476606, 0.9803672032563774, 0.11277778990666365, 0.4821279758062691, 0.7397372080396001, 0.6241593745007569, 0.25949518457715653, 0.021428395696865044, 0.11136391276238113, 0.20574125750714176, 0.5436932281611292, 0.8725881536636315, 0.09356422235744091, 0.5696922361554513, 0.4833510431535772, 0.7172723829816511, 0.6744330014564539, 0.7365215969578738, 0.7014620979853307, 0.16635379595460503, 0.7519392524359211, 0.6302802563476766, 0.6465473372064218, 0.2782644168827087, 0.8973241181321748, 0.6147087615224057, 0.11289261977824389, 0.6222065429823632, 0.8876300567284303, 0.9544218309598148, 0.9628474942673947, 0.24523704160982937, 0.20123705605517142, 0.4591428220018163, 0.7674585439585456, 0.43047241765742583, 0.049685052149313576, 0.6314531728222208, 0.3240828473550122, 0.23783115361813242, 0.6319045968798437, 0.021909851835613243, 0.3104383526266448, 0.12883802954744883, 0.013256254431183554, 0.31971100633224303, 0.06379560617332784, 0.3530595205600023, 0.6614964714175496, 0.39235261629561957, 0.9717007513094378, 0.19572202563580188, 0.3749417393376008, 0.28392080552586735, 0.7967431264051101, 0.3739845264702908, 0.954695004901397, 0.8388030619916982, 0.8524411542500709, 0.6531946334307347, 0.622826963999718, 0.775677518596626, 0.6603179242431576, 0.8042522309348943, 0.2918723705701595, 0.3156533523739583, 0.7696741211795823, 0.8889367443007556, 0.8579051617103431, 0.05838635988689106, 0.5614383738124147, 0.10172480417656149, 0.6797987004829988, 0.05088287720300907, 0.7057859400835449, 0.13423137530194673, 0.5210558765292477, 0.475530442696474, 0.49996463051554757, 0.5519538167400053, 0.337811309310876, 0.9066585053742143, 0.7479524835503003, 0.2724792450397578}
  2. RandomUniform(0, 0.99, 256) → allRandomSeeds, {0.6672619764264518, 0.5386326642839299, 0.9871945630570717, 0.8197364530862499, 0.024874198879509677, 0.34312495586190345, 0.8597269388980237, 0.030230984659786776, 0.7262443514627301, 0.7587791106025003, 0.06549422980261524, 0.9844964713076283, 0.90045781154285, 0.003002814332109367, 0.014601794683650995, 0.5402279186641077, 0.7361940677763092, 0.7030708536926314, 0.4426178076216819, 0.5727329002827127, 0.079668768119683, 0.5321111218223142, 0.4188501484836222, 0.9334436489807381, 0.4847819731150081, 0.6448717559485198, 0.9117714424858241, 0.04827780228558169, 0.19055903477102412, 0.0984319228913516, 0.02845099146423984, 0.48461884167551705, 0.10912495672503018, 0.6342189550265981, 0.9259758193294686, 0.6286058784625344, 0.9790443504398889, 0.44983388403981506, 0.37240274539159357, 0.3232811325627724, 0.023889043737528787, 0.9522637690066902, 0.91278226712082, 0.5129377246355215, 0.6543617206587635, 0.8380719788721592, 0.5122302903087574, 0.9397424552150736, 0.2742085637445294, 0.24110985047131614, 0.962896628692326, 0.7053793599478302, 0.26549541841439883, 0.9741808892543019, 0.6849898396670336, 0.8730659362867671, 0.6476014778163567, 0.7179157316518154, 0.7737662939880373, 0.20383968091372343, 0.6865013200034774, 0.7678787132537996, 0.8145763774376409, 0.2354947575405663, 0.15656015528467535, 0.42962927528173045, 0.016684353362579805, 0.17296845601149283, 0.8461592747313916, 0.6910988135182772, 0.11363841967473358, 0.10533475331207501, 0.6909977361758136, 0.2722209889042442, 0.523466099939352, 0.33685684530499627, 0.1448831962422628, 0.9132663157423582, 0.28590488144082155, 0.9477833714598359, 0.08722931898595838, 0.2857832968378522, 0.05625227285107615, 0.774742684053321, 0.32931840468271956, 0.5734371130070018, 0.2362283911292151, 0.500696906681206, 0.6976680137939146, 0.5743631774876266, 0.3414938794383542, 0.7704627729562626, 0.3625562099066323, 0.7819159667783183, 0.14315017977008188, 0.7197882285575339, 0.39291532906133964, 0.6668954003516708, 0.8166731283747771, 0.4738941410804934, 0.8453950510171513, 0.34147136119574567, 0.3262908504743625, 0.9552473631495048, 0.49542562969707726, 0.29310309918731964, 0.2758153339224633, 0.4901269002939238, 0.14157827834221762, 0.1557640333600266, 0.38837211872175986, 0.31927689234351264, 0.05142454041961499, 0.8357335426530372, 0.1373464339276951, 0.6996102810733185, 0.24743791526812195, 0.8601063568060083, 0.39917361474088103, 0.7715344486330966, 0.5452605525494305, 0.03799558072795191, 0.3353506203328675, 0.8500147119231359, 0.10387430162540712, 0.2371700098198641, 0.1962302148675581, 0.7169746457161218, 0.8578700991972108, 0.512566650353206, 0.8627831600028278, 0.7580514216295337, 0.8261011847100089, 0.19775881811883933, 0.054573808727559826, 0.8380822431468772, 0.1385351578256953, 0.3760358259731317, 0.46269261697490427, 0.18699249297217826, 0.05408613750570726, 0.6842641076641544, 0.28532482898429823, 0.5709892173471351, 0.7666136967836451, 0.9101508500529553, 0.051007561306778304, 0.6931328249895823, 0.39411562678031475, 0.6719198628092761, 0.5404377529168081, 0.5124676036000223, 0.8574221206786655, 0.06873556286636943, 0.8819422190419652, 0.10193686603592933, 0.7394148687842904, 0.9504745461093295, 0.8384056001495376, 0.5522902723815816, 0.6639496845118685, 0.296645555988553, 0.5845759935694878, 0.22211164110978548, 0.8852518616088805, 0.6463971626607805, 0.6332279382182948, 0.6029520873637856, 0.9435845698945592, 0.964628830215641, 0.3466610252305911, 0.14334898732518186, 0.601455237735462, 0.48323446217753224, 0.41644710312267225, 0.15624342404201252, 0.31653117283367305, 0.4214589356707119, 0.13921327265469155, 0.18097843986864148, 0.3127759527995492, 0.2852366477664, 0.6895573309511621, 0.07366826766242307, 0.8700983922951195, 0.24220745595411627, 0.7029409950104925, 0.19827348315861887, 0.8850221369299223, 0.2568686395056743, 0.32159160361825895, 0.36519912756487, 0.8205385354164296, 0.5390233657307372, 0.575065949004739, 0.04465333978232753, 0.26239701404521504, 0.9683241511486188, 0.5141906613892943, 0.8283609497364045, 0.8116166802103905, 0.19828624900527814, 0.44259149698575956, 0.1955219276434307, 0.7889695031353403, 0.19465982720963038, 0.05395911062556989, 0.777136168802037, 0.1865899124770418, 0.31550047228975636, 0.6255727850866972, 0.9269870504006199, 0.7047071634996204, 0.8436664848487377, 0.44941310155375874, 0.27768304436744495, 0.40777644741543856, 0.4367978531199067, 0.5114433279109991, 0.5862011639636397, 0.9740012676187216, 0.8950947889489914, 0.19732452662036642, 0.6053597401114987, 0.37539403052433157, 0.23341565996205632, 0.09895266449805476, 0.6955664374512684, 0.44211356774328714, 0.8713523826741955, 0.24076604458550008, 0.8950459733720844, 0.3248475028898973, 0.6730172848131547, 0.09777018180725792, 0.13654621111351625, 0.08247451774814446, 0.25728992729480443, 0.3586620653186078, 0.08425449446907467, 0.5391451541079877, 0.9792227122024386, 0.24730932051781773, 0.506324780382547, 0.7152626417282522, 0.07419299243261566, 0.13415000782651193, 0.6260249832436288, 0.17062346027211361, 0.7882161480256384, 0.7158270749807243, 0.43805796998225244, 0.6051947592527388, 0.35059690383656306, 0.5765567525702273, 0.7960919213305689}
  3. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  4. Zip(round(A), A, Join[{Take[allData, 1 + Mod[countQuestion, 250]], Take[allData, 1, 1 + Mod[countQuestion, 250]]}]) → revisedData
  5. Sequence(Sort[Sequence[revisedData(B + A), A, 0, randomChildren - 1]], B, 1, (randomDataAmount * randomChildren), randomChildren) → currentData
  6. Sequence(Sum[Zip[If[A ≟ 0, "B", "G"], A, currentData(B)]], B, 1, Length[currentData]) → sequencedPossibilities
  7. Sum(Sequence[If[n ≤ randomBoy, "B", "G"], n, 1, randomChildren]) → targetText
  8. CountIf(n ≟ targetText, n, sequencedPossibilities) → numTarget
  9. Corner(5) → layoutAppSize
  10. Polygon({(-0.5, -0.5), (-0.5, 6.7), (12, 6.7), (12, -6.5)}) → viewbox
  11. onboardingTextOffset(Corner[onboardingText, 1], Corner[onboardingText, 3], onboardingCase, Corner[1], Corner[3], Corner[5]) → OnboardingTextOffset
  12. Max(Frequency[sequencedPossibilities]) → dataHeight
  13. DotPlot(sequencedPossibilities) → targetDotPlot
  14. Unique(sequencedPossibilities) → combinationOptions
  15. Length(combinationOptions) → dataWidth
  16. Zip(((x(n) * xMult), (y(n) * yMult)) - (shift, 0), n, targetDotPlot) → scaledDotPlot
  17. Sequence(((12 * n / dataWidth) - shift, 0), n, 1, dataWidth) → PointStarts
  18. Zip(Text[RotateText[n, 60°], k - (0.85, 0), true, true], n, combinationOptions, k, PointStarts) → dotText
  19. Sequence(Segment[(0, (yMult * n)), (-0.1, (yMult * n))], n, 1, dataHeight) → dotVSeg
  20. Zip(Segment[n, n - (0, 0.1)], n, PointStarts) → dotHSeg
  21. Sequence(Text[If[dataHeight ≤ 10 ∨ Mod[n, 2] ≟ 0, n + "", ""], (If[n < 10, -0.8, -1], (yMult * n) + 0.25), true, true], n, 1, dataHeight) → dotNums
  22. Sequence(Segment[(0, ((yMult * n) * 4)), (12, ((yMult * n) * 4))], n, 1, dataHeight / 4) → hLines
  23. Textfield(input1) → InputBox1
  24. Numerator(correctInput) → numSimp
  25. Segment((0, 0), (0, 6.7)) → ySeg
  26. VerticalText("Frequency") → yLabel

AI Drawing Prompt Reference

Create a statistical chart for probability simulation. Generate a large set of random numbers as simulation data, then select a subset as the current sample. Convert each value in the sample to a label of 'B' or 'G' based on whether it is below a threshold, forming multiple possible combination sequences. Count the frequency of each combination and display the distribution using a dot plot. Below the plot, label all possible combination options, and on the left, show frequency numbers. Draw axes and grid lines. Also, include an input box for entering an expected probability value and display its simplified numerator.
The prompt above was compiled from the original drawing commands and is for reference only.