中奖概率与期望值：多选与开放题互动练习

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2025-05-14 17:22

所属专题 Laplace-Wahrscheinlichkeit

该GeoGebra范例围绕概率与期望值设计了两类互动练习。第一类为多选题，通过一个包含1个主奖、24个安慰奖和75个空奖的100张彩票的抽奖情境，让学生判断关于中奖概率的多个陈述是否正确，例如主奖概率小于空奖概率、安慰奖概率等于0.24等。第二类为两道开放题：一道比较三个不同彩票游戏（A、B、C）的中奖概率，要求学生用概率论证哪个游戏的中奖机会最大；另一道涉及每张3欧元、共100张彩票的期望收益计算，并判断游戏是否公平。核心数学概念包括拉普拉斯概率（等可能事件概率）、概率比较、期望值及公平性判断。作图方面，该范例使用了GeoGebra的If、Sequence、Element、Textfield、Join、Text、SelectedIndex、Translate、PolyLine等命令，并包含多个文本输入框、按钮（如btnBack、btnNew、btnNext）、反馈文本（txtFeedba

GeoGebra 命令及参数

If(step ≟ 2, correct02, step ≟ 3, correct03, step ≟ 4, correct04, step ≟ 5, correct05, false) → correct
Sequence(Random[1, 10], i, 1, 10) → randomNumbers, {9, 6, 2, 8, 1, 6, 10, 10, 4, 4}
Element(randomNumbers, Mod[questionCount, Length[randomNumbers]] + 1) → randomNumber
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, {{(-0.3900000000000001, 10.379999999999999), (0.6121097046413506, 10.379999999999999), (1.6142194092827014, 10.379999999999999), (2.6163291139240523, 10.379999999999999), (3.618438818565403, 10.379999999999999), (4.620548523206754, 10.379999999999999), (5.622658227848104, 10.379999999999999), (6.624767932489455, 10.379999999999999), (7.626877637130805, 10.379999999999999), (8.628987341772156, 10.379999999999999), (9.631097046413508, 10.379999999999999), (10.633206751054859, 10.379999999999999), (11.63531645569621, 10.379999999999999), (12.63742616033756, 10.379999999999999), (13.63953586497891, 10.379999999999999), (14.64164556962026, 10.379999999999999), (15.643755274261611, 10.379999999999999), (16.645864978902964, 10.379999999999999), (17.647974683544312, 10.379999999999999), (18.650084388185665, 10.379999999999999)}, {(-0.3900000000000001, 9.377134670487106), (0.6121097046413506, 9.377134670487106), (1.6142194092827014, 9.377134670487106), (2.6163291139240523, 9.377134670487106), (3.618438818565403, 9.377134670487106), (4.620548523206754, 9.377134670487106), (5.622658227848104, 9.377134670487106), (6.624767932489455, 9.377134670487106), (7.626877637130805, 9.377134670487106), (8.628987341772156, 9.377134670487106), (9.631097046413508, 9.377134670487106), (10.633206751054859, 9.377134670487106), (11.63531645569621, 9.377134670487106), (12.63742616033756, 9.377134670487106), (13.63953586497891, 9.377134670487106), (14.64164556962026, 9.377134670487106), (15.643755274261611, 9.377134670487106), (16.645864978902964, 9.377134670487106), (17.647974683544312, 9.377134670487106), (18.650084388185665, 9.377134670487106)}, {(-0.3900000000000001, 8.374269340974212), (0.6121097046413506, 8.374269340974212), (1.6142194092827014, 8.374269340974212), (2.6163291139240523, 8.374269340974212), (3.618438818565403, 8.374269340974212), (4.620548523206754, 8.374269340974212), (5.622658227848104, 8.374269340974212), (6.624767932489455, 8.374269340974212), (7.626877637130805, 8.374269340974212), (8.628987341772156, 8.374269340974212), (9.631097046413508, 8.374269340974212), (10.633206751054859, 8.374269340974212), (11.63531645569621, 8.374269340974212), (12.63742616033756, 8.374269340974212), (13.63953586497891, 8.374269340974212), (14.64164556962026, 8.374269340974212), (15.643755274261611, 8.374269340974212), (16.645864978902964, 8.374269340974212), (17.647974683544312, 8.374269340974212), (18.650084388185665, 8.374269340974212)}, {(-0.3900000000000001, 7.371404011461318), (0.6121097046413506, 7.371404011461318), (1.6142194092827014, 7.371404011461318), (2.6163291139240523, 7.371404011461318), (3.618438818565403, 7.371404011461318), (4.620548523206754, 7.371404011461318), (5.622658227848104, 7.371404011461318), (6.624767932489455, 7.371404011461318), (7.626877637130805, 7.371404011461318), (8.628987341772156, 7.371404011461318), (9.631097046413508, 7.371404011461318), (10.633206751054859, 7.371404011461318), (11.63531645569621, 7.371404011461318), (12.63742616033756, 7.371404011461318), (13.63953586497891, 7.371404011461318), (14.64164556962026, 7.371404011461318), (15.643755274261611, 7.371404011461318), (16.645864978902964, 7.371404011461318), (17.647974683544312, 7.371404011461318), (18.650084388185665, 7.371404011461318)}, {(-0.3900000000000001, 6.368538681948425), (0.6121097046413506, 6.368538681948425), (1.6142194092827014, 6.368538681948425), (2.6163291139240523, 6.368538681948425), (3.618438818565403, 6.368538681948425), (4.620548523206754, 6.368538681948425), (5.622658227848104, 6.368538681948425), (6.624767932489455, 6.368538681948425), (7.626877637130805, 6.368538681948425), (8.628987341772156, 6.368538681948425), (9.631097046413508, 6.368538681948425), (10.633206751054859, 6.368538681948425), (11.63531645569621, 6.368538681948425), (12.63742616033756, 6.368538681948425), (13.63953586497891, 6.368538681948425), (14.64164556962026, 6.368538681948425), (15.643755274261611, 6.368538681948425), (16.645864978902964, 6.368538681948425), (17.647974683544312, 6.368538681948425), (18.650084388185665, 6.368538681948425)}, {(-0.3900000000000001, 5.365673352435532), (0.6121097046413506, 5.365673352435532), (1.6142194092827014, 5.365673352435532), (2.6163291139240523, 5.365673352435532), (3.618438818565403, 5.365673352435532), (4.620548523206754, 5.365673352435532), (5.622658227848104, 5.365673352435532), (6.624767932489455, 5.365673352435532), (7.626877637130805, 5.365673352435532), (8.628987341772156, 5.365673352435532), (9.631097046413508, 5.365673352435532), (10.633206751054859, 5.365673352435532), (11.63531645569621, 5.365673352435532), (12.63742616033756, 5.365673352435532), (13.63953586497891, 5.365673352435532), (14.64164556962026, 5.365673352435532), (15.643755274261611, 5.365673352435532), (16.645864978902964, 5.365673352435532), (17.647974683544312, 5.365673352435532), (18.650084388185665, 5.365673352435532)}, {(-0.3900000000000001, 4.362808022922637), (0.6121097046413506, 4.362808022922637), (1.6142194092827014, 4.362808022922637), (2.6163291139240523, 4.362808022922637), (3.618438818565403, 4.362808022922637), (4.620548523206754, 4.362808022922637), (5.622658227848104, 4.362808022922637), (6.624767932489455, 4.362808022922637), (7.626877637130805, 4.362808022922637), (8.628987341772156, 4.362808022922637), (9.631097046413508, 4.362808022922637), (10.633206751054859, 4.362808022922637), (11.63531645569621, 4.362808022922637), (12.63742616033756, 4.362808022922637), (13.63953586497891, 4.362808022922637), (14.64164556962026, 4.362808022922637), (15.643755274261611, 4.362808022922637), (16.645864978902964, 4.362808022922637), (17.647974683544312, 4.362808022922637), (18.650084388185665, 4.362808022922637)}, {(-0.3900000000000001, 3.359942693409744), (0.6121097046413506, 3.359942693409744), (1.6142194092827014, 3.359942693409744), (2.6163291139240523, 3.359942693409744), (3.618438818565403, 3.359942693409744), (4.620548523206754, 3.359942693409744), (5.622658227848104, 3.359942693409744), (6.624767932489455, 3.359942693409744), (7.626877637130805, 3.359942693409744), (8.628987341772156, 3.359942693409744), (9.631097046413508, 3.359942693409744), (10.633206751054859, 3.359942693409744), (11.63531645569621, 3.359942693409744), (12.63742616033756, 3.359942693409744), (13.63953586497891, 3.359942693409744), (14.64164556962026, 3.359942693409744), (15.643755274261611, 3.359942693409744), (16.645864978902964, 3.359942693409744), (17.647974683544312, 3.359942693409744), (18.650084388185665, 3.359942693409744)}, {(-0.3900000000000001, 2.3570773638968507), (0.6121097046413506, 2.3570773638968507), (1.6142194092827014, 2.3570773638968507), (2.6163291139240523, 2.3570773638968507), (3.618438818565403, 2.3570773638968507), (4.620548523206754, 2.3570773638968507), (5.622658227848104, 2.3570773638968507), (6.624767932489455, 2.3570773638968507), (7.626877637130805, 2.3570773638968507), (8.628987341772156, 2.3570773638968507), (9.631097046413508, 2.3570773638968507), (10.633206751054859, 2.3570773638968507), (11.63531645569621, 2.3570773638968507), (12.63742616033756, 2.3570773638968507), (13.63953586497891, 2.3570773638968507), (14.64164556962026, 2.3570773638968507), (15.643755274261611, 2.3570773638968507), (16.645864978902964, 2.3570773638968507), (17.647974683544312, 2.3570773638968507), (18.650084388185665, 2.3570773638968507)}, {(-0.3900000000000001, 1.3542120343839574), (0.6121097046413506, 1.3542120343839574), (1.6142194092827014, 1.3542120343839574), (2.6163291139240523, 1.3542120343839574), (3.618438818565403, 1.3542120343839574), (4.620548523206754, 1.3542120343839574), (5.622658227848104, 1.3542120343839574), (6.624767932489455, 1.3542120343839574), (7.626877637130805, 1.3542120343839574), (8.628987341772156, 1.3542120343839574), (9.631097046413508, 1.3542120343839574), (10.633206751054859, 1.3542120343839574), (11.63531645569621, 1.3542120343839574), (12.63742616033756, 1.3542120343839574), (13.63953586497891, 1.3542120343839574), (14.64164556962026, 1.3542120343839574), (15.643755274261611, 1.3542120343839574), (16.645864978902964, 1.3542120343839574), (17.647974683544312, 1.3542120343839574), (18.650084388185665, 1.3542120343839574)}, {(-0.3900000000000001, 0.3513467048710641), (0.6121097046413506, 0.3513467048710641), (1.6142194092827014, 0.3513467048710641), (2.6163291139240523, 0.3513467048710641), (3.618438818565403, 0.3513467048710641), (4.620548523206754, 0.3513467048710641), (5.622658227848104, 0.3513467048710641), (6.624767932489455, 0.3513467048710641), (7.626877637130805, 0.3513467048710641), (8.628987341772156, 0.3513467048710641), (9.631097046413508, 0.3513467048710641), (10.633206751054859, 0.3513467048710641), (11.63531645569621, 0.3513467048710641), (12.63742616033756, 0.3513467048710641), (13.63953586497891, 0.3513467048710641), (14.64164556962026, 0.3513467048710641), (15.643755274261611, 0.3513467048710641), (16.645864978902964, 0.3513467048710641), (17.647974683544312, 0.3513467048710641), (18.650084388185665, 0.3513467048710641)}, {(-0.3900000000000001, -0.6515186246418292), (0.6121097046413506, -0.6515186246418292), (1.6142194092827014, -0.6515186246418292), (2.6163291139240523, -0.6515186246418292), (3.618438818565403, -0.6515186246418292), (4.620548523206754, -0.6515186246418292), (5.622658227848104, -0.6515186246418292), (6.624767932489455, -0.6515186246418292), (7.626877637130805, -0.6515186246418292), (8.628987341772156, -0.6515186246418292), (9.631097046413508, -0.6515186246418292), (10.633206751054859, -0.6515186246418292), (11.63531645569621, -0.6515186246418292), (12.63742616033756, -0.6515186246418292), (13.63953586497891, -0.6515186246418292), (14.64164556962026, -0.6515186246418292), (15.643755274261611, -0.6515186246418292), (16.645864978902964, -0.6515186246418292), (17.647974683544312, -0.6515186246418292), (18.650084388185665, -0.6515186246418292)}, {(-0.3900000000000001, -1.6543839541547243), (0.6121097046413506, -1.6543839541547243), (1.6142194092827014, -1.6543839541547243), (2.6163291139240523, -1.6543839541547243), (3.618438818565403, -1.6543839541547243), (4.620548523206754, -1.6543839541547243), (5.622658227848104, -1.6543839541547243), (6.624767932489455, -1.6543839541547243), (7.626877637130805, -1.6543839541547243), (8.628987341772156, -1.6543839541547243), (9.631097046413508, -1.6543839541547243), (10.633206751054859, -1.6543839541547243), (11.63531645569621, -1.6543839541547243), (12.63742616033756, -1.6543839541547243), (13.63953586497891, -1.6543839541547243), (14.64164556962026, -1.6543839541547243), (15.643755274261611, -1.6543839541547243), (16.645864978902964, -1.6543839541547243), (17.647974683544312, -1.6543839541547243), (18.650084388185665, -1.6543839541547243)}, {(-0.3900000000000001, -2.6572492836676176), (0.6121097046413506, -2.6572492836676176), (1.6142194092827014, -2.6572492836676176), (2.6163291139240523, -2.6572492836676176), (3.618438818565403, -2.6572492836676176), (4.620548523206754, -2.6572492836676176), (5.622658227848104, -2.6572492836676176), (6.624767932489455, -2.6572492836676176), (7.626877637130805, -2.6572492836676176), (8.628987341772156, -2.6572492836676176), (9.631097046413508, -2.6572492836676176), (10.633206751054859, -2.6572492836676176), (11.63531645569621, -2.6572492836676176), (12.63742616033756, -2.6572492836676176), (13.63953586497891, -2.6572492836676176), (14.64164556962026, -2.6572492836676176), (15.643755274261611, -2.6572492836676176), (16.645864978902964, -2.6572492836676176), (17.647974683544312, -2.6572492836676176), (18.650084388185665, -2.6572492836676176)}, {(-0.3900000000000001, -3.660114613180511), (0.6121097046413506, -3.660114613180511), (1.6142194092827014, -3.660114613180511), (2.6163291139240523, -3.660114613180511), (3.618438818565403, -3.660114613180511), (4.620548523206754, -3.660114613180511), (5.622658227848104, -3.660114613180511), (6.624767932489455, -3.660114613180511), (7.626877637130805, -3.660114613180511), (8.628987341772156, -3.660114613180511), (9.631097046413508, -3.660114613180511), (10.633206751054859, -3.660114613180511), (11.63531645569621, -3.660114613180511), (12.63742616033756, -3.660114613180511), (13.63953586497891, -3.660114613180511), (14.64164556962026, -3.660114613180511), (15.643755274261611, -3.660114613180511), (16.645864978902964, -3.660114613180511), (17.647974683544312, -3.660114613180511), (18.650084388185665, -3.660114613180511)}}
Textfield(input021) → InputBox021
Textfield(input022) → InputBox022
Textfield(input031) → InputBox031
Textfield(input032) → InputBox032
Textfield(input041) → InputBox041
Textfield(input042) → InputBox042
Textfield(input051) → InputBox051
Textfield(input052) → InputBox052
Textfield(input061) → InputBox061
Textfield(input062) → InputBox062
Textfield(input063) → InputBox063
Textfield(input064) → InputBox064
Textfield(input065) → InputBox065
If(step ≟ 1, (-0.05, 6.5), step ≟ 6 ∨ step ≟ 6.25 ∨ step ≟ 6.5, (-0.05, 9.35), (-0.05, 7.75)) → PosTable
If(step < 2, 3, 4) → AnzahlZeilen
Sequence(Sequence[PosTable + (Sum[ListSpalte, k], (-Sum[ListZeile, m])), k, 0, AnzahlSpalten], m, 0, AnzahlZeilen) → TablePosMat, {{(-0.05, 6.5), (2.6500000000000004, 6.5), (5.6000000000000005, 6.5), (8.55, 6.5), (11.100000000000001, 6.5), (15.05, 6.5)}, {(-0.05, 5.6), (2.6500000000000004, 5.6), (5.6000000000000005, 5.6), (8.55, 5.6), (11.100000000000001, 5.6), (15.05, 5.6)}, {(-0.05, 4.7), (2.6500000000000004, 4.7), (5.6000000000000005, 4.7), (8.55, 4.7), (11.100000000000001, 4.7), (15.05, 4.7)}, {(-0.05, 3.8), (2.6500000000000004, 3.8), (5.6000000000000005, 3.8), (8.55, 3.8), (11.100000000000001, 3.8), (15.05, 3.8)}}
Text("\textsf{Losart\vphantom{g}}", Midpoint[TablePosMat(1, 1), TablePosMat(2, 1)] + (0, 0.05), false, true, 1, 0) → CapTable11
Text("\textsf{Anzahl\vphantom{g}}", Midpoint[TablePosMat(2, 1), TablePosMat(3, 1)] + (0, 0.05), false, true, 1, 0) → CapTable21
Text("\textsf{Gewinn\vphantom{g}}", Midpoint[TablePosMat(3, 1), TablePosMat(4, 1)] + (0, 0.05), false, true, 1, 0) → CapTable31
Text("\textsf{mit Smiley\vphantom{g}}", Midpoint[TablePosMat(1, 2), TablePosMat(2, 2)] + (0, 0.05), false, true, 1, 0) → CapTable12
Text("\textsf{mit Sonne\vphantom{g}}", Midpoint[TablePosMat(1, 3), TablePosMat(2, 3)] + (0, 0.05), false, true, 1, 0) → CapTable13
Text("\textsf{mit Hai\vphantom{g}}", Midpoint[TablePosMat(1, 4), TablePosMat(2, 4)] + (0, 0.05), false, true, 1, 0) → CapTable14
Text("\textsf{mit Regenbogen\vphantom{g}}", Midpoint[TablePosMat(1, 5), TablePosMat(2, 5)] + (0, 0.05), false, true, 1, 0) → CapTable15
Text("\textsf{70\vphantom{g}}", Midpoint[TablePosMat(2, 2), TablePosMat(3, 2)] + (0, 0.05), false, true, 1, 0) → CapTable22
Text("\textsf{20\vphantom{g}}", Midpoint[TablePosMat(2, 3), TablePosMat(3, 3)] + (0, 0.05), false, true, 1, 0) → CapTable23
Text("\textsf{9\vphantom{g}}", Midpoint[TablePosMat(2, 4), TablePosMat(3, 4)] + (0, 0.05), false, true, 1, 0) → CapTable24
Text("\textsf{1\vphantom{g}}", Midpoint[TablePosMat(2, 5), TablePosMat(3, 5)] + (0, 0.05), false, true, 1, 0) → CapTable25
Text("\textsf{0 €\vphantom{g}}", Midpoint[TablePosMat(3, 2), TablePosMat(4, 2)] + (0, 0.05), false, true, 1, 0) → CapTable32
Text("\textsf{5 €\vphantom{g}}", Midpoint[TablePosMat(3, 3), TablePosMat(4, 3)] + (0, 0.05), false, true, 1, 0) → CapTable33
Text("\textsf{10 €\vphantom{g}}", Midpoint[TablePosMat(3, 4), TablePosMat(4, 4)] + (0, 0.05), false, true, 1, 0) → CapTable34
Text("\textsf{100 €\vphantom{g}}", Midpoint[TablePosMat(3, 5), TablePosMat(4, 5)] + (0, 0.05), false, true, 1, 0) → CapTable35
Text("\textsf{\frac{70}{100}\vphantom{g}}", Midpoint[TablePosMat(4, 2), TablePosMat(5, 2)] + (0, 0.05), false, true, 1, 0) → CapTable42
Text("\textsf{\frac{20}{100}\vphantom{g}}", Midpoint[TablePosMat(4, 3), TablePosMat(5, 3)] + (0, 0.05), false, true, 1, 0) → CapTable43
Text("\textsf{\frac{9}{100}\vphantom{g}}", Midpoint[TablePosMat(4, 4), TablePosMat(5, 4)] + (0, 0.05), false, true, 1, 0) → CapTable44
Text("\textsf{\frac{1}{100}\vphantom{g}}", Midpoint[TablePosMat(4, 5), TablePosMat(5, 5)] + (0, 0.05), false, true, 1, 0) → CapTable45
SelectedIndex(dropdown071) → index071
Translate(imgFace, Vector[PosImgFace1, Corner[CapIB02, 1] + (3.7, 0.2)]) → imgFaceW
Translate(imgSun, Vector[PosImgSun1, Corner[CapIB02, 1] + (3.7, 0.2)]) → imgSunW
Translate(imgShark, Vector[PosImgShark1, Corner[CapIB02, 1] + (3.7, 0.2)]) → imgSharkW
Translate(imgRainbow, Vector[PosImgRainbow1, Corner[CapIB02, 1] + (3.7, 0.2)]) → imgRainbowW
Translate(imgFace, Vector[PosImgFace1, Corner[txt063, 4] + (1.45, -0.7)]) → imgFaceE
Textfield(input066) → InputBox066
Textfield(input067) → InputBox067
If(step ≟ 7, (0.5, -1.57), step ≟ 6, (0.1, 6), (0.5, -1)) → PosImgMichael1
PolyLine(PosSB, Corner[txt073, 1] + (-0.1, -0.1) + (0, 0.5), Corner[txt073, 4] + (-0.1, 0.1), Corner[txt073, 3] + (0.1, 0.1), Corner[txt073, 2] + (0.1, -0.1), Corner[txt073, 1] + (-0.1, -0.1) + (0.5, 0), PosSB) → sb07
Textfield(input071) → InputBox071
Textfield(input072) → InputBox072
If(step < 6, Corner[CapIB02, 3] + (1, -0.25), step ≤ 6.5, (8, -0.75), (9, 0)) → PosBtnSolution
PolyLine(PosSB, Corner[txt052, 1] + (0.25, -0.1), Corner[txt052, 2] + (0.1, -0.1), Corner[txt052, 3] + (0.1, 0.1), Corner[txt052, 4] + (-0.1, 0.1), Corner[txt052, 1] + (-0.1, 0.25), PosSB) → sb05
Translate(imgRainbow, Vector[PosImgRainbow1, Corner[txt063, 4] + (14.3, -0.7)]) → imgRainbowE
Translate(imgSun, Vector[PosImgSun1, Corner[txt063, 4] + (5.62, -0.7)]) → imgSunE
Translate(imgShark, Vector[PosImgShark1, Corner[txt063, 4] + (9.8, -0.7)]) → imgSharkE
PolyLine(PosSB, Corner[txt067, 1] + (0.25, -0.1), Corner[txt067, 2] + (0.1, -0.1), Corner[txt067, 3] + (0.1, 0.1), Corner[txt067, 4] + (-0.1, 0.1), Corner[txt067, 1] + (-0.1, 0.25), PosSB) → sb06
Text("\textsf{Wahrschein-\\lichkeit W\vphantom{g}}", Midpoint[TablePosMat(4, 1), TablePosMat(5, 1)] + (0, 0), false, true, 1, 0) → CapTable41