Galton Board Simulator
Drop balls through a pegboard and watch the binomial distribution converge to the normal curve, with a chi-squared test against the theoretical bin counts.
Last reviewed by the Radiatus Cloud team
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The bell curve is a count of paths, not a law of nature
A ball passing n rows of pegs makes n independent left-or-right choices, and the number landing in bin k is the number of paths reaching it, which is n choose k. Those binomial coefficients form Pascal's triangle, and Pascal's triangle looks like a bell curve because the middle rows have vastly more paths than the edges. Nothing about physics or measurement produces the shape; it comes from counting arrangements, and that is the whole content of the central limit theorem in one apparatus.
Convergence is fast and the tails are where it is slowest
Even ten rows produce a shape hard to distinguish from normal by eye near the centre, while the extreme bins stay noticeably discrete much longer. That pattern is general: sums of independent variables approach normality quickly in the middle and slowly in the tails, which is exactly why risk models built on normal assumptions fail on rare events rather than typical ones. The apparatus makes the discrepancy visible because the tail bins are individually countable.
Biasing the pegs shifts the peak without breaking the shape
Changing the probability of going right from one half to some other value moves the peak to n times p and narrows or widens the curve, and it stays approximately normal. The distribution is only symmetric when p is one half, and the skew at extreme values is another place where the normal approximation degrades, which is why the rule of thumb requires n times p and n times one minus p both to exceed about ten.
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Frequently Asked Questions
Why does a bell curve appear?
Because the number of balls in each bin is the number of paths reaching it, which is a binomial coefficient. Pascal’s triangle has vastly more paths through the middle than the edges, and that count is the shape.
How many rows are needed?
Ten already looks convincingly normal near the centre. The tails take far longer to converge, which is the general pattern for sums of independent variables.
What does the chi-squared test show?
Whether the observed counts differ from the theoretical binomial more than sampling noise explains. A large value with many balls means the simulation is biased; a large value with few balls usually means too few balls.
What happens if the pegs are biased?
The peak moves to n times p and the curve skews. It stays roughly normal while n times p and n times one minus p both exceed about ten, which is where the usual rule of thumb comes from.
Is this really how the central limit theorem works?
It is one clean instance of it: a sum of independent identical binary choices approaching a normal shape. The theorem is far more general, and the apparatus shows why the result is about counting rather than about any particular process.
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How to Use
Set the rows and ball count, then drop them.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.