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Non-transitive Spinners Probability Simulation and Comparison

Non-transitive Spinners Probability Simulation and Comparison
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2013-01-06 05:06
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This GeoGebra example demonstrates the counterintuitive phenomenon of non-transitive spinners, where three spinners (red, green, blue) exhibit cyclic winning advantages across repeated trials, resembling rock-paper-scissors dynamics. Core mathematical concepts include probability and statistics, random experiments, expected value, proportional comparison, and non-transitive relations. The construction uses GeoGebra commands such as Textfield, Element, Circle, Sequence, Vector, Rotate, Segment, Polygon, and LaTeX, together with objects like rand, listr/listg/listb, inputBoxR/G/B, trials, θ_R/G/B, n_R/G/B, C_R/G/B, c_R/G/B, P_R/G/B to create spinner sectors, rotation angles, trial-count input boxes, and win-rate statistics lists. It is suitable for secondary-school probability and statistics units, classroom inquiry, after-class simulation experiments, and intuitive demonstrations of non-transitive dice or spinner principles.

GeoGebra Commands & Parameters

  1. Textfield(listr) → inputBoxR
  2. Textfield(listg) → inputBoxG
  3. Textfield(listb) → inputBoxB
  4. Textfield(trials) → inputBox1
  5. Element(listr, round(θ_R / 60 + 0.5)) → n_R
  6. Element(listg, round(θ_G / 60 + 0.5)) → n_G
  7. Element(listb, round(θ_B / 60 + 0.5)) → n_B
  8. Circle(C_R, 2) → c_R
  9. Circle(C_G, 2) → c_G
  10. Circle(C_B, 2) → c_B
  11. Sequence(Rotate[P_R, ((k * 60))°, C_R], k, 0, 5) → listrpt
  12. Sequence(CircleSector[C_R, Element[listrpt, Mod[k, 6] + 1], Element[listrpt, Mod[k + 1, 6] + 1]], k, 0, 5) → listrseg
  13. Element(listrseg, 1) → rseg1
  14. Element(listrseg, 2) → rseg_2
  15. Element(listrseg, 3) → rseg_3
  16. Element(listrseg, 4) → rseg_4
  17. Element(listrseg, 5) → rseg_5
  18. Element(listrseg, 6) → rseg_6
  19. Sequence(Rotate[P_G, ((k * 60))°, C_G], k, 0, 5) → listgpt
  20. Sequence(CircleSector[C_G, Element[listgpt, Mod[k, 6] + 1], Element[listgpt, Mod[k + 1, 6] + 1]], k, 0, 5) → listgseg
  21. Element(listgseg, 1) → gseg_1
  22. Element(listgseg, 2) → gseg_2
  23. Element(listgseg, 3) → gseg_3
  24. Element(listgseg, 4) → gseg_4
  25. Element(listgseg, 5) → gseg_5
  26. Element(listgseg, 6) → gseg_6
  27. Sequence(Rotate[P_B, ((k * 60))°, C_B], k, 0, 5) → listbpt
  28. Sequence(CircleSector[C_B, Element[listbpt, Mod[k, 6] + 1], Element[listbpt, Mod[k + 1, 6] + 1]], k, 0, 5) → listbseg
  29. Element(listbseg, 1) → bseg_1
  30. Element(listbseg, 2) → bseg_2
  31. Element(listbseg, 3) → bseg_3
  32. Element(listbseg, 4) → bseg_4
  33. Element(listbseg, 5) → bseg_5
  34. Element(listbseg, 6) → bseg_6
  35. Vector(C_R, C_R + (1.5, 0)) → v_R
  36. Vector(C_G, C_G + (1.5, 0)) → v_G
  37. Vector(C_B, C_B + (1.5, 0)) → v_B
  38. Rotate(v_R, θ_R°, C_R) → arrow_R
  39. Rotate(v_G, θ_G°, C_G) → arrow_G
  40. Rotate(v_B, θ_B°, C_B) → arrow_B
  41. Segment((0, 0), (0, 5.5)) → yaxis_1
  42. Polygon(Xrb_5, Xrb_6, Xrb_6 + (0, (5 * ebr / count)), Xrb_5 + (0, (5 * ebr / count))) → polyebr, a, b, c, d
  43. LaTeX(bg, true, true) → text1
  44. LaTeX(br, true, true) → text2
  45. LaTeX(ebr, true, true) → text3
  46. LaTeX(egb, true, true) → text4
  47. LaTeX(erg, true, true) → text5
  48. LaTeX(gb, true, true) → text6
  49. LaTeX(gr, true, true) → text7
  50. LaTeX(rb, true, true) → text8
  51. LaTeX(rg, true, true) → text9
  52. Polygon(Xrb_1, Xrb_2, Xrb_2 + (0, (5 * rb / count)), Xrb_1 + (0, (5 * rb / count))) → polyrb, e, f, i, j
  53. Polygon(Xrb_3, Xrb_4, Xrb_4 + (0, (5 * br / count)), Xrb_3 + (0, (5 * br / count))) → polybr, g, h, k, l
  54. Segment((-8, 8), (-8, 13.5)) → yaxis_2
  55. Segment((8, 8), (13, 8)) → xaxis_3
  56. Segment((8, 8), (8, 13.5)) → yaxis_3
  57. Segment((-8, 8), (-3, 8)) → xaxis_2
  58. Segment((0, 0), (5, 0)) → xaxis_1
  59. Polygon(Xbg_5, Xbg_6, Xbg_6 + (0, (5 * egb / count)), Xbg_5 + (0, (5 * egb / count))) → polyegb, m, n, p, q
  60. Polygon(Xbg_1, Xbg_2, Xbg_2 + (0, (5 * bg / count)), Xbg_1 + (0, (5 * bg / count))) → polybg, xbg_1, xbg_2, r, s
  61. Polygon(Xbg_3, Xbg_4, Xbg_4 + (0, (5 * gb / count)), Xbg_3 + (0, (5 * gb / count))) → polygb, t, a_1, b_1, c_1
  62. Polygon(Xgr_5, Xgr_6, Xgr_6 + (0, (5 * erg / count)), Xgr_5 + (0, (5 * erg / count))) → polyerg, xgr_5, xgr_6, d_1, e_1
  63. Polygon(Xgr_1, Xgr_2, Xgr_2 + (0, (5 * gr / count)), Xgr_1 + (0, (5 * gr / count))) → polygr, xgr_1, xgr_2, f_1, g_1
  64. Polygon(Xgr_3, Xgr_4, Xgr_4 + (0, (5 * rg / count)), Xgr_3 + (0, (5 * rg / count))) → polyrg, xgr_3, xgr_4, h_1, i_1