The Bell-Walk

Four distributions, one bell. The middle is the average. Every question is just: how much energy does it take to get from the middle to where you're standing?

The bell comes first. Before any formula, the bell already exists — it emerged from independent forces piling up without coordination. Thick in the middle, thin at the edges. The force first. The formula after.

How to know which one you're in

Binomial — count of tries + percentage per try

Hypergeometric — exact counts in a finite pool, no percentage

Poisson — average rate over a window, no fixed count

Normal — told directly ("normalfordelt") or summing many independent things

Choose the distribution

The Bell

Binomial Bell

Bin(10, 0.6) — 11 doors from 0 to 10. Standing at k = 6.

Connected Spells

Drainage feeds the counting. The nCr inside binomial and hypergeometric is the same Drainage from Kap 3.

Probability House builds the mass. The probabilities here live in the same house of cells from Kap 1+2.

Weighing Table measures the spread. Variance and standard deviation from the Weighing Table (Kap 5+6) feed into the normal distribution’s z-score.

Formula Reference — Kap 7

Binomial — Bin(n, p)Kap 7.1–7.3

P(X = k) = C(n,k) · p^k · (1−p)^n−k

E[X] = np Var[X] = np(1−p) Patterns × success-energy × failure-energy

Hypergeometric — Hyp(N, M, n)Kap 7.4

P(X = k) = C(M,k) · C(N−M, n−k) / C(N, n)

E[X] = nM/N Gunstige over mulige. Three Drainages — glowing pool, dark pool, all hands.

Poisson — Po(λ)Kap 7.5

P(X = k) = λ^k · e^−λ / k!

E[X] = λ Var[X] = λ Energy pushes out, void pulls back, factorial cleans up.

Normal — N(μ, σ²)Kap 7.6–7.7

z = (X − μ) / σ → z-table

68% within ±1σ, 95% within ±2σ, 99.7% within ±3σ. Distance from center in stretches.