Galton Board, explained.
A Galton board builds a binomial distribution by accumulating independent left–right choices across successive rows of pegs.
Why it happens
Many paths can reach a central bin, while far fewer paths reach an extreme. Counting those paths produces the exact binomial probabilities; with more rows its standardized shape approaches a normal distribution.
Sums of many independent binary steps produce a binomial distribution whose shape approaches a bell curve as the number of steps increases.
Read the result
Drop two hundred balls and compare observed bins with exact expected counts. Resample to distinguish the distribution’s shape from one finite batch. The simulation counts right turns rather than modeling physical collisions.
A worked example
Eight rows of choices
Every path has eight independent fair steps, so there are 256 equally likely paths.
Seventy paths contain exactly four right turns, giving central probability 70/256.
Two hundred balls therefore have expected central count about 54.69, but an observed count can be above or below that number.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Each ball takes n independent fair steps. The count of right steps chooses its bin. Observed counts are compared with exact binomial probabilities times 200.
Where this idea is useful
A practical use
A roughly bell-shaped total can arise from adding many small independent contributions, even when each contribution is not bell-shaped.
A common misconception
“The exact expected count must be a whole observed count.”
Expectation averages over possible batches and can be fractional; a particular batch contains an integer number of balls.
What this explanation leaves out
- Independence and equal step probabilities are imposed. A bell shape in real data does not by itself identify its cause.
Does every collection of small influences produce a bell curve?
No. Independence, distributional conditions and scaling matter; heavy tails or dependence can change the result.
Are your observed totals built from independent contributions, or do those contributions share a cause?
Associated thinkers
Further reading
Explore the original research or the teaching reference behind this experiment.