Every arrangement you see is one sample drawn from an infinite distribution of possible arrangements.
The Shape of Uncertainty
Probability is not about predicting what will happen. It is about describing the space of what could happen, and how likely each possibility is. The bell curve is not a prediction -- it is a map of ignorance, rendered beautiful by the central limit theorem.
Given enough independent samples from any distribution, their mean converges to a Gaussian. This is not a hypothesis. It is a mathematical certainty about uncertainty itself.
Bayes' Theorem
Prior beliefs, updated by evidence, yield posterior knowledge. The theorem is an epistemology: it describes how rational minds should change.
P(H|E) = P(E|H) · P(H) / P(E)
Galton Board
Variance
The average squared distance from the mean. Variance is the width of the bell, the uncertainty that remains after expectation.
Var(X) = E[(X−μ)²]
Expected Value
The weighted average of all possible outcomes. The expected value of a fair die is 3.5 -- a number the die can never show.
Convergence
The law of large numbers promises that as trials increase, the sample mean approaches the true mean. Flip a coin once and you have chaos. Flip it a thousand times and you have certainty. The beauty of probability lies in this transformation from the unpredictable to the inevitable.
Monte Carlo
When analysis fails, simulation succeeds. Throw enough random darts at a circle inscribed in a square and you can compute pi. Named for the casino where chance is the house specialty.
Independence
Two events are independent when knowing one tells you nothing about the other. The coin does not remember its last flip.
In the limit, randomness reveals its architecture. What seemed chaotic at small scales resolves into elegant structure at scale. This is the promise of probability: patience yields pattern.
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