彩色球团掷骰：拉普拉斯概率与相对频率预测练习

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2025-05-14 20:55

所属专题 Laplace-Wahrscheinlichkeit

该范例演示在“彩色球团/随机抽取”这一单步随机实验中，如何计算拉普拉斯概率，并把它解释为相对频率的预测值。核心数学概念包括：单步随机试验、等可能性假设下的拉普拉斯概率、将概率作为相对频率的预测、以及对选项答案的验证与反馈。作图与交互构建要点来自构造摘要：使用 Join 将对象集合起来；用 Element 获取被选元素；用 If 与 SelectedIndex 根据用户选择切换反馈；通过 Textfield 输入并显示学生答案；用 Sequence 生成题目/随机选项序列，并结合 RemoveUndefined 清理未定义项；用 Text 与 Corner 设置提示/说明文本位置；通过按钮对象（btnValidate、btnNew、btnTryAgain、showsolution 等）控制流程；用 validate、showHint、txtHintTypical 提供常见思路提示；并用 re

GeoGebra 命令及参数

Join({{8}, Shuffle[Remove[1…numberOfQuestions, {8}]]}) → randomQuestions
Element(randomQuestions, Mod[questionCount - 1, Length[randomQuestions]] + 1) → randomQuestion
If(randomQuestion ∈ {2, 4}, option2 ∧ (¬option1), option1 ∧ (¬option2)) → correctMC
SelectedIndex(dropdown) → indexDD
If(randomQuestion < 7, correctMC, randomQuestion ∈ {7, 8, 9}, correctIB ∧ correctBruch ∧ If[randomQuestion ≟ 9, correct3, true], correctDD) → correct
Textfield(input) → inputBox
Join({Sequence[Segment[Corner[4] + (RasterPixMat * ((k * RasterStepXY), 0)), Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY))], k, 0, RasterMaxX / RasterStepXY], Sequence[Segment[Corner[4] + (RasterPixMat * (0, (k * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], k, 0, RasterMaxY / RasterStepXY]}) → RasterStrecken
Join({Sequence[Text[UnicodeToLetter[65 + k], Midpoint[Corner[4] + (RasterPixMat * ((k * RasterStepXY), RasterMaxY)), Corner[4] + (RasterPixMat * (((k + 1) * RasterStepXY), RasterMaxY))], false, true, 0, -1], k, 0, RasterMaxX / RasterStepXY - 1], Sequence[Text[("" * k), Midpoint[Corner[4] + (RasterPixMat * (RasterMaxX, ((k - 1) * RasterStepXY))), Corner[4] + (RasterPixMat * (RasterMaxX, (k * RasterStepXY)))], false, true, 1, 0], k, 1, RasterMaxY / RasterStepXY]}) → RasterBeschriftung
Sequence(Sequence[Corner[4] + (RasterPixMat * ((k * RasterStepXY), (m * RasterStepXY))), k, 0, RasterMaxX / RasterStepXY], m, 0, RasterMaxY / RasterStepXY) → RasterPunkte, {{(-14.020000000000012, 8.279999999999994), (-13.01749373433585, 8.279999999999994), (-12.01498746867169, 8.279999999999994), (-11.012481203007528, 8.279999999999994), (-10.009974937343365, 8.279999999999994), (-9.007468671679204, 8.279999999999994), (-8.004962406015043, 8.279999999999994), (-7.002456140350881, 8.279999999999994), (-5.99994987468672, 8.279999999999994), (-4.997443609022557, 8.279999999999994), (-3.994937343358396, 8.279999999999994), (-2.992431077694235, 8.279999999999994), (-1.9899248120300737, 8.279999999999994), (-0.9874185463659124, 8.279999999999994), (0.01508771929825059, 8.279999999999994), (1.0175939849624118, 8.279999999999994), (2.0201002506265713, 8.279999999999994)}, {(-14.020000000000012, 7.2766555183946435), (-13.01749373433585, 7.2766555183946435), (-12.01498746867169, 7.2766555183946435), (-11.012481203007528, 7.2766555183946435), (-10.009974937343365, 7.2766555183946435), (-9.007468671679204, 7.2766555183946435), (-8.004962406015043, 7.2766555183946435), (-7.002456140350881, 7.2766555183946435), (-5.99994987468672, 7.2766555183946435), (-4.997443609022557, 7.2766555183946435), (-3.994937343358396, 7.2766555183946435), (-2.992431077694235, 7.2766555183946435), (-1.9899248120300737, 7.2766555183946435), (-0.9874185463659124, 7.2766555183946435), (0.01508771929825059, 7.2766555183946435), (1.0175939849624118, 7.2766555183946435), (2.0201002506265713, 7.2766555183946435)}, {(-14.020000000000012, 6.273311036789293), (-13.01749373433585, 6.273311036789293), (-12.01498746867169, 6.273311036789293), (-11.012481203007528, 6.273311036789293), (-10.009974937343365, 6.273311036789293), (-9.007468671679204, 6.273311036789293), (-8.004962406015043, 6.273311036789293), (-7.002456140350881, 6.273311036789293), (-5.99994987468672, 6.273311036789293), (-4.997443609022557, 6.273311036789293), (-3.994937343358396, 6.273311036789293), (-2.992431077694235, 6.273311036789293), (-1.9899248120300737, 6.273311036789293), (-0.9874185463659124, 6.273311036789293), (0.01508771929825059, 6.273311036789293), (1.0175939849624118, 6.273311036789293), (2.0201002506265713, 6.273311036789293)}, {(-14.020000000000012, 5.2699665551839425), (-13.01749373433585, 5.2699665551839425), (-12.01498746867169, 5.2699665551839425), (-11.012481203007528, 5.2699665551839425), (-10.009974937343365, 5.2699665551839425), (-9.007468671679204, 5.2699665551839425), (-8.004962406015043, 5.2699665551839425), (-7.002456140350881, 5.2699665551839425), (-5.99994987468672, 5.2699665551839425), (-4.997443609022557, 5.2699665551839425), (-3.994937343358396, 5.2699665551839425), (-2.992431077694235, 5.2699665551839425), (-1.9899248120300737, 5.2699665551839425), (-0.9874185463659124, 5.2699665551839425), (0.01508771929825059, 5.2699665551839425), (1.0175939849624118, 5.2699665551839425), (2.0201002506265713, 5.2699665551839425)}, {(-14.020000000000012, 4.266622073578592), (-13.01749373433585, 4.266622073578592), (-12.01498746867169, 4.266622073578592), (-11.012481203007528, 4.266622073578592), (-10.009974937343365, 4.266622073578592), (-9.007468671679204, 4.266622073578592), (-8.004962406015043, 4.266622073578592), (-7.002456140350881, 4.266622073578592), (-5.99994987468672, 4.266622073578592), (-4.997443609022557, 4.266622073578592), (-3.994937343358396, 4.266622073578592), (-2.992431077694235, 4.266622073578592), (-1.9899248120300737, 4.266622073578592), (-0.9874185463659124, 4.266622073578592), (0.01508771929825059, 4.266622073578592), (1.0175939849624118, 4.266622073578592), (2.0201002506265713, 4.266622073578592)}, {(-14.020000000000012, 3.2632775919732415), (-13.01749373433585, 3.2632775919732415), (-12.01498746867169, 3.2632775919732415), (-11.012481203007528, 3.2632775919732415), (-10.009974937343365, 3.2632775919732415), (-9.007468671679204, 3.2632775919732415), (-8.004962406015043, 3.2632775919732415), (-7.002456140350881, 3.2632775919732415), (-5.99994987468672, 3.2632775919732415), (-4.997443609022557, 3.2632775919732415), (-3.994937343358396, 3.2632775919732415), (-2.992431077694235, 3.2632775919732415), (-1.9899248120300737, 3.2632775919732415), (-0.9874185463659124, 3.2632775919732415), (0.01508771929825059, 3.2632775919732415), (1.0175939849624118, 3.2632775919732415), (2.0201002506265713, 3.2632775919732415)}, {(-14.020000000000012, 2.259933110367891), (-13.01749373433585, 2.259933110367891), (-12.01498746867169, 2.259933110367891), (-11.012481203007528, 2.259933110367891), (-10.009974937343365, 2.259933110367891), (-9.007468671679204, 2.259933110367891), (-8.004962406015043, 2.259933110367891), (-7.002456140350881, 2.259933110367891), (-5.99994987468672, 2.259933110367891), (-4.997443609022557, 2.259933110367891), (-3.994937343358396, 2.259933110367891), (-2.992431077694235, 2.259933110367891), (-1.9899248120300737, 2.259933110367891), (-0.9874185463659124, 2.259933110367891), (0.01508771929825059, 2.259933110367891), (1.0175939849624118, 2.259933110367891), (2.0201002506265713, 2.259933110367891)}, {(-14.020000000000012, 1.2565886287625405), (-13.01749373433585, 1.2565886287625405), (-12.01498746867169, 1.2565886287625405), (-11.012481203007528, 1.2565886287625405), (-10.009974937343365, 1.2565886287625405), (-9.007468671679204, 1.2565886287625405), (-8.004962406015043, 1.2565886287625405), (-7.002456140350881, 1.2565886287625405), (-5.99994987468672, 1.2565886287625405), (-4.997443609022557, 1.2565886287625405), (-3.994937343358396, 1.2565886287625405), (-2.992431077694235, 1.2565886287625405), (-1.9899248120300737, 1.2565886287625405), (-0.9874185463659124, 1.2565886287625405), (0.01508771929825059, 1.2565886287625405), (1.0175939849624118, 1.2565886287625405), (2.0201002506265713, 1.2565886287625405)}, {(-14.020000000000012, 0.25324414715719), (-13.01749373433585, 0.25324414715719), (-12.01498746867169, 0.25324414715719), (-11.012481203007528, 0.25324414715719), (-10.009974937343365, 0.25324414715719), (-9.007468671679204, 0.25324414715719), (-8.004962406015043, 0.25324414715719), (-7.002456140350881, 0.25324414715719), (-5.99994987468672, 0.25324414715719), (-4.997443609022557, 0.25324414715719), (-3.994937343358396, 0.25324414715719), (-2.992431077694235, 0.25324414715719), (-1.9899248120300737, 0.25324414715719), (-0.9874185463659124, 0.25324414715719), (0.01508771929825059, 0.25324414715719), (1.0175939849624118, 0.25324414715719), (2.0201002506265713, 0.25324414715719)}, {(-14.020000000000012, -0.7501003344481596), (-13.01749373433585, -0.7501003344481596), (-12.01498746867169, -0.7501003344481596), (-11.012481203007528, -0.7501003344481596), (-10.009974937343365, -0.7501003344481596), (-9.007468671679204, -0.7501003344481596), (-8.004962406015043, -0.7501003344481596), (-7.002456140350881, -0.7501003344481596), (-5.99994987468672, -0.7501003344481596), (-4.997443609022557, -0.7501003344481596), (-3.994937343358396, -0.7501003344481596), (-2.992431077694235, -0.7501003344481596), (-1.9899248120300737, -0.7501003344481596), (-0.9874185463659124, -0.7501003344481596), (0.01508771929825059, -0.7501003344481596), (1.0175939849624118, -0.7501003344481596), (2.0201002506265713, -0.7501003344481596)}, {(-14.020000000000012, -1.753444816053511), (-13.01749373433585, -1.753444816053511), (-12.01498746867169, -1.753444816053511), (-11.012481203007528, -1.753444816053511), (-10.009974937343365, -1.753444816053511), (-9.007468671679204, -1.753444816053511), (-8.004962406015043, -1.753444816053511), (-7.002456140350881, -1.753444816053511), (-5.99994987468672, -1.753444816053511), (-4.997443609022557, -1.753444816053511), (-3.994937343358396, -1.753444816053511), (-2.992431077694235, -1.753444816053511), (-1.9899248120300737, -1.753444816053511), (-0.9874185463659124, -1.753444816053511), (0.01508771929825059, -1.753444816053511), (1.0175939849624118, -1.753444816053511), (2.0201002506265713, -1.753444816053511)}, {(-14.020000000000012, -2.7567892976588624), (-13.01749373433585, -2.7567892976588624), (-12.01498746867169, -2.7567892976588624), (-11.012481203007528, -2.7567892976588624), (-10.009974937343365, -2.7567892976588624), (-9.007468671679204, -2.7567892976588624), (-8.004962406015043, -2.7567892976588624), (-7.002456140350881, -2.7567892976588624), (-5.99994987468672, -2.7567892976588624), (-4.997443609022557, -2.7567892976588624), (-3.994937343358396, -2.7567892976588624), (-2.992431077694235, -2.7567892976588624), (-1.9899248120300737, -2.7567892976588624), (-0.9874185463659124, -2.7567892976588624), (0.01508771929825059, -2.7567892976588624), (1.0175939849624118, -2.7567892976588624), (2.0201002506265713, -2.7567892976588624)}, {(-14.020000000000012, -3.760133779264212), (-13.01749373433585, -3.760133779264212), (-12.01498746867169, -3.760133779264212), (-11.012481203007528, -3.760133779264212), (-10.009974937343365, -3.760133779264212), (-9.007468671679204, -3.760133779264212), (-8.004962406015043, -3.760133779264212), (-7.002456140350881, -3.760133779264212), (-5.99994987468672, -3.760133779264212), (-4.997443609022557, -3.760133779264212), (-3.994937343358396, -3.760133779264212), (-2.992431077694235, -3.760133779264212), (-1.9899248120300737, -3.760133779264212), (-0.9874185463659124, -3.760133779264212), (0.01508771929825059, -3.760133779264212), (1.0175939849624118, -3.760133779264212), (2.0201002506265713, -3.760133779264212)}}
RemoveUndefined({RO1}) → ListeRichtig
RemoveUndefined({FO1}) → ListeFalsch
Join({ListeRichtig, ListeFalsch}) → Reihenfolge
Text(Reihenfolge(1), Corner[4] + (pixMat * (PosStart + (36, 16) + (((1 - 1) * disPixX), ((1 - 1) * disPixY)))), false, true, 1, 0) → txtO1
Text(Reihenfolge(2), Corner[4] + (pixMat * (PosStart + (36, 16) + (((2 - 1) * disPixX), ((2 - 1) * disPixY)))), false, true, 1, 0) → txtO2
Sequence(Circle[Points01(k), radiusK], k, 1, Length[Points01]) → Kugel01
Sequence(Circle[Points02(k), radiusK], k, 1, Length[Points02]) → Kugel02
Sequence(Circle[Points03(k), radiusK], k, 1, Length[Points03]) → Kugel03
Sequence(Circle[Points04(k), radiusK], k, 1, Length[Points04]) → Kugel04
Sequence(Circle[Points05(k), radiusK], k, 1, Length[Points05]) → Kugel05
Sequence(Circle[Points06(k), radiusK], k, 1, Length[Points06]) → Kugel06
Join({Kugel01, Kugel02, Kugel03, Kugel04, Kugel05, Kugel06}) → KugelRand
Text((("\textsf{" * If[randomQuestion ∈ {1, 3, 5, 6}, "✓ Richtig", "✘ Falsch"]) * "}"), Midpoint[Vertex[polyMC1(1), 2], Vertex[polyMC1(1), 3]] + (0.5, 0.05), false, true, 1, 0) → txtFB1
Text((("\textsf{" * If[randomQuestion ∈ {2, 4}, "✓ Richtig", "✘ Falsch"]) * "}"), Midpoint[Vertex[polyMC2(1), 2], Vertex[polyMC2(1), 3]] + (0.5, 0.05), false, true, 1, 0) → txtFB2
Textfield(input1) → InputBox1
Textfield(input2) → InputBox2
Textfield(input3) → InputBox3
Corner(txtIB, 1) → CornerIB
If(randomQuestion < 7, option1 ∨ option2, randomQuestion ∈ {7, 8, 9}, Defined[input] ∧ Defined[input1] ∧ Defined[input2] ∧ If[randomQuestion ≟ 9, Defined[input3], true], indexDD ≠ 1) → clickableBtnValidate
If(randomQuestion < 7, (-13.5, -1.25), randomQuestion ∈ {7, 8, 9}, (-8, 2.85) + If[randomQuestion ≟ 9, (0, -1.25), (0, 0)], (-2.5, 1.25)) → PosBtnValidate
If(randomQuestion ≥ 10, (-13.68, 0.5), randomQuestion ∈ {7, 8}, (-13.68, 0.5), (-13.68, 0.5 - 1.2)) → PosSolution
If(randomQuestion < 10, PosBtnValidate + (0, 0), PosBtnValidate + (-3.75, 0.25)) → PosFB