# Probability Lab

Canonical: https://manyspace.aporia.systems/tools/probability
Updated: 2026-10-13

One flip tells you nothing. A thousand flips tell you p. Flip the coin and watch luck average out.

## Try this

- Feel the lie of small n. Press “Flip 1” five separate times (reset between). Note how different five single flips feel from one run of five — the single flip is all story, no statistics.
- Watch the pin. Set p = 0.70, press “+1k”, and watch the curve dive from chaos onto the dashed line. The first 30 flips are drama; the last 900 are destiny.
- Chase the gap. Open “Deeper: seeds & replay”, fix a seed, and add flips until |p̂ − p| drops under 0.01. Then shuffle the seed and notice: different path, same destination.

## Why repetition tames luck

One coin flip is pure noise: heads or tails, nothing learned. But the running average — heads so far divided by flips so far — has a destiny. As flips pile up, the wild early swings shrink and the curve gets pinned near the true chance p. That pinning is the law of large numbers: averages of many independent trials converge to the expected value.

The chart above shows the law happening. Early on, three heads in a row can park the average at 1.00 — a lie the next flips dismantle. By flip 1,000 the average barely budges, and its typical wobble falls roughly like 1/√n. Quadrupling your flips roughly halves the wobble. Randomness does not disappear; it averages out.

## Related tools

- [Derivative Visualizer](https://manyspace.aporia.systems/tools/derivative)
- [Integral Explorer](https://manyspace.aporia.systems/tools/integral)
- [Unit Circle Explorer](https://manyspace.aporia.systems/tools/unit-circle)
- [Complex Plane Explorer](https://manyspace.aporia.systems/tools/complex-plane)
- [Bayes Lab](https://manyspace.aporia.systems/tools/bayes)
- [Eigen Explorer](https://manyspace.aporia.systems/tools/eigenvectors)
- [Fourier Series](https://manyspace.aporia.systems/tools/fourier-series)