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Galton Board & Binomial Distribution by Helmholtz - class 10a

Galton Board & Binomial Distribution by Helmholtz - class 10a
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2016-10-08 20:42
Probability & StatisticsGeometric Construction
This example demonstrates the relationship between the Galton Board experiment results and the Binomial Distribution, visually showing the probability pattern of random events under repeated trials. Core math concepts include Binomial Distribution, Probability & Statistics, Random Events, Expected Value, and Median Value. Construction highlights involve commands like Sequence, Join, Shuffle, Append, Point, PolyLine, and Sum, along with objects such as n, p, board, h, and more. Suitable for high school students and above learning probability statistics, discrete mathematics, or university-level introductory probability courses.

GeoGebra Commands & Parameters

  1. Sequence(Sequence[(i, j), i, j / 2, n - j / 2], j, 1, n) → board
  2. Sequence(Polygon[(i - 0.5, 0), (i + 0.5, 0), (i + 0.5, (BinomialDist[n, 1 - p, n - i, false] * h)), (i - 0.5, (BinomialDist[n, 1 - p, n - i, false] * h))], i, 0, n) → BinomialDist
  3. Join({Sequence[-0.5, i, 1, ((10 * n) * (1 - p))], Sequence[0.5, i, 1, ((10 * p) * n)]}) → unshuffle
  4. Shuffle(unshuffle) → shuffle
  5. Append((n / 2, n + 1), Sequence[(n / 2, n + 1) + (Sum[First[shuffle, i]], (-i)), i, 1, n]) → path
  6. Sequence(Segment[(i - 1 / 2, 0), (i - 1 / 2, 1 / 2)], i, 0, n + 1) → container
  7. Append(Sequence[Segment[Element[path, i - 1], Element[path, i]], i, 1, n + 1], Segment[Element[path, n + 1], Element[path, n + 1] + (0, -1)]) → line
  8. Point(line, t) → P
  9. PolyLine(((n - 1) / 2, n + 1), ((n - 1) / 2, n + 0.5), (-0.5, 0.5), (-0.5, 0), (n + 0.5, 0), (n + 0.5, 0.5), ((n - 1) / 2 + 1, n + 0.5), ((n + 1) / 2, n + 1)) → l
  10. Sequence(Polygon[(i - 1.5, 0), (i - 0.5, 0), (i - 0.5, (Element[rec, i] / Sum[rec] * h)), (i - 1.5, (Element[rec, i] / Sum[rec] * h))], i, 1, n + 1) → box_1
  11. Sum(rec) → number
  12. Sequence(Text[(Element[BinomialDist, i] / h * 100) + "%", (i - 1.4, -0.4)], i, 1, n + 1) → binopercent
  13. Sequence(Text[Element[rec, i], (i - 1.4, -0.9)], i, 1, n + 1) → list1
  14. Sequence(Text[(Element[rec, i] / number * 100) + "%", (i - 1.4, -1.4)], i, 1, n + 1) → list2
  15. Sequence(Text[("box " * i), (i - 0.3, 0.25)], i, 0, n) → boxnumber