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Calculating Probability Using a Pinball Simulation

Calculating Probability Using a Pinball Simulation
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2023-02-21 09:45
Probability & StatisticsGeometric ConstructionSlider Animation
This example demonstrates probability by simulating a pinball's movement to compare theoretical and experimental probabilities. The key mathematical concept is geometric probability, where the ratio of areas is used. Construction involves GeoGebra commands like RandomUniform, Line, and Segment to create the path and regions. Suitable for grades 6-8 to explore probability in geometry.

GeoGebra Commands & Parameters

  1. RandomUniform(0, 0.99, 200) → allRandomSeeds, {0.785694756253163, 0.9806807303442829, 0.4744696787349965, 0.07302956912855027, 0.6002911955890478, 0.41685885562467967, 0.6861535088471415, 0.047571139460948306, 0.3954983311529015, 0.7097217494407272, 0.24800847363370232, 0.2201887275198625, 0.8106600284811379, 0.5363199874340305, 0.5253245889388841, 0.5246633761588722, 0.2684459430993198, 0.15901160380693782, 0.8667465453496013, 0.8024028540295866, 0.5334640264145052, 0.9530060467445283, 0.45942020793128985, 0.8930432233486214, 0.356136941967206, 0.4730930985799898, 0.03231489525644536, 0.03427510264778363, 0.4900763763991845, 0.9214011190905512, 0.08245423274434464, 0.17263588028993462, 0.7944204085717418, 0.1524329391380891, 0.8462389362157464, 0.014383224618356555, 0.35177010880420334, 0.2494697045735961, 0.9864338778301837, 0.631374254747728, 0.6974681000705967, 0.7377123568218829, 0.5747174479578188, 0.48112456830770295, 0.9570073319406002, 0.05424704507426261, 0.8099557658651568, 0.16290525668751152, 0.5353403914783431, 0.09307487583335723, 0.5935644933956055, 0.7728600196182621, 0.8947799112038506, 0.4150390218217557, 0.7654605276350136, 0.9606849640006532, 0.1170374688490024, 0.8575103835551244, 0.8871813716352823, 0.4984111973212997, 0.38974418209889955, 0.8371607062709006, 0.04276399973916673, 0.36826662980233515, 0.2885278415926325, 0.5010557766321515, 0.14292612970015536, 0.5614657229443936, 0.9223915377994091, 0.7766436393187238, 0.23398160721307087, 0.3230869059872952, 0.8627466581400911, 0.9499913671869552, 0.9497473620563281, 0.4899454180414229, 0.15499719289750144, 0.6938415340248048, 0.6024636135148606, 0.778554525500967, 0.4160555752506634, 0.7527130928520013, 0.05744988634133021, 0.1488917247792147, 0.79158947294477, 0.39136377614034673, 0.9058579356164059, 0.15674630906194814, 0.8631568495785614, 0.3254599436486467, 0.33045925304250096, 0.30292347558341715, 0.7676815257567239, 0.5492667937899818, 0.02091809600380783, 0.39151523299042396, 0.16995335016046006, 0.5745764944987947, 0.09745238985940329, 0.7018214036667263, 0.7230079294747065, 0.7479326779873388, 0.6451733855073069, 0.5093150624508108, 0.37134718256829985, 0.18949021456121806, 0.9857213901295837, 0.5502940849936204, 0.6080081302378343, 0.38656497043081506, 0.8980781965514182, 0.562546500669413, 0.8884685570048507, 0.11808308075105671, 0.3366380779753214, 0.53048034522324, 0.09229904431357008, 0.2427311109017028, 0.4217582719680741, 0.35772915928134114, 0.8473277017273857, 0.22363729202771557, 0.8760857030800236, 0.8432422247225062, 0.8608791189671421, 0.6858814385828887, 0.3189702167493783, 0.4383044995215036, 0.688381476205186, 0.100034899476585, 0.2712261132311666, 0.871803822858032, 0.22130561896066775, 0.7252741879877513, 0.27339245307116583, 0.15937404203784403, 0.5340071534728592, 0.42939072833354924, 0.6724683813586252, 0.3787901810435957, 0.3425879566634997, 0.3402363811935522, 0.4048638553228518, 0.8091777811354688, 0.9483754907779045, 0.5237985471231233, 0.37700699179482017, 0.415221311845558, 0.27403857450203734, 0.8681902562229618, 0.8096880315685927, 0.19961732988815506, 0.09171531457403852, 0.8103702214868501, 0.28779633697338197, 0.3875157327770149, 0.5098242198587452, 0.453374757088205, 0.8936198643550596, 0.04562857702831169, 0.3284617168304894, 0.4562470481370943, 0.6693287220355588, 0.846058368672597, 0.6124591727811451, 0.6166920572648316, 0.8526518421024316, 0.18070859595805502, 0.966280140341301, 0.3777637072399701, 0.5790832850317251, 0.026387527950223585, 0.355983185473702, 0.654608531468472, 0.9742172734387733, 0.4334012401169554, 0.5804584868852238, 0.3454438000042206, 0.7210128162810492, 0.1415555289848067, 0.2470658340787871, 0.38254005161261584, 0.6189275100302891, 0.5470464674681815, 0.9456594890398617, 0.9240072991386645, 0.8149356411662082, 0.07171770402843557, 0.49977993107852225, 0.30565114721404335, 0.9704603956608897, 0.652597806210376, 0.8845254426202807, 0.44406888633110203, 0.6431999998667703, 0.8445045413039075, 0.7572856914476114, 0.4274978987478695, 0.048717295771883884, 0.10181277581867074}
  2. Take(allRandomSeeds, randIndex, randIndex + numRandoms - 1) → currentRandomSeeds
  3. RemoveUndefined(Zip[If[n, m], n, chosenSegs, m, boolPoss]) → boolSuccess
  4. IndexOf(true, boolSegs) → chosenInd
  5. Line((5.35, -1.9), (5.85, -1.85)) → guideLine2
  6. Angle(guideLine2, xAxis) → β
  7. Segment((5.35, -1.9), (5.85, -1.85)) → pinballBottom
  8. Intersect(pinballBottom, guideLine) → PullSeg1
  9. Intersect(y = -3, guideLine) → PullSeg2
  10. Segment(PullSeg1, PullSeg2) → pullSegNew
  11. Point(pullSegNew) → PullPointA
  12. Zip(Text[n, k - (lettWidth / 2, ((-lettHeight)) / 2 + 0.4) + sizeVect, true, true], n, {"A", "B", "C", "D"}, k, Points) → pointLabels
  13. Segment((5.6, -2.14), (5.6, -3)) → pullSeg
  14. Sum(Zip[If[n, m, 0], n, chosenSegs, m, numSegs]) → top
  15. Polygon(Zip[PixelsToPoint[Corner[1], Corner[3], Corner[5], A], A, {(pixelsLeft, pixelsTop), (pixelsLeft, y(Corner[5]) - pixelsBottom), Corner[5] - (pixelsRight, pixelsBottom), (x(Corner[5]) - pixelsRight, pixelsTop)}]) → borderBox
  16. Corner(5) → layoutAppSize
  17. Segment((-4, 0), (4, 0)) → segmentStart
  18. Sequence(Midpoint[Points(n), Points(n + 1)], n, 1, 3) → MidSegs
  19. First(textSteps, If[step ≟ 1, 1, step - 1]) → textCurrentSteps
  20. Zip(Text[n, k - (numWidth / 2, -0.85) - numVect, true, true], n, {randomNum2, randomNum1 - (randomNum2 + randomNum3), randomNum3}, k, MidSegs) → textSegs
  21. Flatten({Sequence[Midpoint[MidSegs(n), MidSegs(n + 1)], n, 1, 2], (4.8, 0)}) → SignMids
  22. Zip(Text[n, k - (numWidth / 2, -0.85) - numVect, true, true], n, {"+", "+", "=\hspace{0.5cm}" + randomNum1}, k, SignMids) → textPlusStep2
  23. Segment(Points(numSegUse1), Points(numSegUse1 + 2)) → segmentProb
  24. CurveCartesian((0.25 * sin(t)) + 5.6, (-0.25 * ((curvFunct(yDiff) * t / ((pi * 6))) + cos(t) + 7.55)), t, 0, ((2 * pi) * 6)) → cur
  25. Angle((0, 0), Intersect[guideLine, xAxis], PullSeg1) → α
  26. Rotate(cur, (-(pi / 2 - α))) → cur'
  27. Translate(cur', (-0.23, -0.6)) → cur''
  28. Rotate(Polygon[{PullPointA + (0, 1 / ((2 * sqrt(3)))), PullPointA + (-1 / 4, -1 / ((4 * sqrt(3)))), PullPointA + (1 / 4, -1 / ((4 * sqrt(3))))}], (-β), PullPointA) → pullTriangle
  29. IntegralBetween(pinBallBTop, pinBallBottom, 4.2, 5.3) → shading
  30. IntegralBetween(pinBallBTop, guideLine2, 5.3, 5.8) → shading2
  31. Line(PullSeg1, PullSeg2) → pullLine
  32. Segment(PointArrow1, PointArrow2) → arrowSeg
  33. Segment(Points(1), Points(2)) → segment1
  34. Segment(Points(2), Points(3)) → segment2
  35. Segment(Points(3), Points(4)) → segment3
  36. Zip(Text[n, k - (numWidth / 2, -0.85) - numVect, true, true], n, Zip[If[n, m, ""], n, chosenSegs, m, numSegs], k, MidSegs) → textSegs3
  37. If(Defined[chosenInd] ∧ Length[chosenSignMids(chosenInd)] > 0, Sequence["+", n, 1, Length[chosenSignMids(chosenInd)]], {}) → plusSeq
  38. Zip(Text[n, k - (numWidth / 2, -0.85) - numVect, true, true], n, If[Length[plusSeq] > 0, Flatten[{plusSeq, "=\hspace{0.5cm}" + top}], {"=\hspace{0.5cm}" + top}], k, Flatten[{If[Defined[chosenInd], chosenSignMids(chosenInd), {}], Element[SignMids, 3]}]) → textPlusStep3

AI Drawing Prompt Reference

Draw a geometric simulation of a pinball's movement to compare theoretical and experimental probabilities. Start by drawing a horizontal line as the starting point, then draw a slanted line segment as the pinball path. Next, draw two vertical line segments as the starting and ending points of the pinball. Randomly select several points on the path to draw several line segments representing the pinball's movement. Calculate and label angles and segment lengths, and use shading to indicate probabilities in different areas. Finally, draw a triangle to indicate the direction of the pinball's movement and label the probabilities on the path.
The prompt above was compiled from the original drawing commands and is for reference only.