확률 — Probability

The Architecture of Uncertainty

Where marble colonnades dissolve into probability clouds and classical certainty yields to the beauty of the stochastic.

Explore the probability space

II

Distributions

“The theory of probabilities is at bottom nothing but common sense reduced to calculus.” — Laplace

Gaussian

f(x) = (1/σ√2π) e^{-(x-μ)²/2σ²}

The bell curve — nature’s most ubiquitous probability distribution. From measurement errors to human heights, the Gaussian emerges wherever many small independent effects combine.

Poisson

P(k) = λ^ke^-λ / k!

The distribution of rare events in vast spaces of possibility. How many stars appear in a patch of sky? How many calls arrive per minute? Poisson knows.

Bernoulli

P(X=1) = p, P(X=0) = 1-p

The fundamental binary — success or failure, heads or tails, being or nothingness. Every complex probability begins with this elemental choice.

III

Convergence

“In the long run, we are all converging to the mean.”

The Law of Large Numbers

As the number of trials increases without bound, the sample mean converges to the expected value. Chaos at the individual level resolves into deterministic certainty at scale — the profound paradox at the heart of probability theory.

Simulated trials 0

Running mean —

IV

Axioms of Probability

Kolmogorov’s foundation — three axioms upon which all of probability rests.

1

Non-negativity

P(A) ≥ 0

The probability of any event is a non-negative real number. There is no such thing as negative likelihood — only degrees of possibility stretching from impossibility to certainty.

2

Unitarity

P(Ω) = 1

The probability of the entire sample space is one. Something must happen. The universe of outcomes is complete and exhaustive — a closed marble hall where every possibility echoes.

3

Additivity

P(A ∪ B) = P(A) + P(B)

For mutually exclusive events, the probability of their union equals the sum of their individual probabilities. Disjoint possibilities combine without interference.

In the marble halls of probability, every outcome whispers its likelihood into the void.

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