Chance makes patterns: sources & model

Release a few virtual balls, then a thousand. Change the chance of a right turn and compare the results with a mathematical prediction.

What supports the explanation?

What this model assumes

1. Each row is an independent Bernoulli choice. A real pegboard can have correlated impacts, bounce dynamics and uneven geometry; those are not modeled.
2. The seed is deterministic, and changing it chooses another repeatable run. This does not establish that a physical coin is fair.
3. No gambling advice, forecasts or inferential confidence intervals. Long-run convergence does not mean each next trial compensates for past outcomes.
4. Count all the paths: For n independent choices, P(X = k) = C(n,k)pᵏ(1 − p)ⁿ⁻ᵏ. C(n,k) counts the possible orders of k right turns. Probabilities sum to one. At n = 4 and p = 0.5, bin probabilities are 1, 4, 6, 4, 1 divided by 16.
5. Center and spread: E[X] = np and Var(X) = np(1 − p). The expected number in bin k is sample size × P(X = k). Expected need not mean most likely, and an expected count can be fractional.
6. Randomness you can replay: A versioned 32-bit pseudorandom generator supplies uniform draws, compared with p. A fixed seed and configuration reproduce the same choices. This is a mathematical Bernoulli board, not a physical collision simulation or cryptographic randomness.

What has been checked

Analytical reference cases, conservation or transition invariants, finite drawing commands, bounded setup parsing, discovery and route integrity are checked automatically. These checks do not establish anatomical fidelity, learner outcomes or browser/device compatibility. Independent subject review, learner trials, comprehensive accessibility review and browser video encoding checks remain pending.

Each source supports the associated claim. Sources do not certify this implementation or its visuals.