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確率 — the mathematics of maybe

Foundations

What is Probability?

The measure of the likelihood that an event will occur. A number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

P(A) = n(A)n(S)
Roll the die
P = 1/6
Bayes’ Theorem
P(A|B) = P(B|A) · P(A)P(B)

Updating beliefs in the light of new evidence. The foundation of statistical inference.

Classic Problem

The Urn Problem

Draw without replacement. How does each draw reshape the odds?

Decision Trees 0.6 0.4 0.3 0.7 0.5 0.5 0.18 0.42 0.20 0.20

“Probability is the very guide of life.”

— Cicero
The Bell Curve μ
Convergence

Law of Large Numbers

As trials increase, the sample mean converges to the expected value. Certainty emerges from chaos.

n → ∞
1654

The Birth of Probability

A correspondence between Blaise Pascal and Pierre de Fermat over the “problem of points” laid the foundation for the mathematical theory of probability. What began as a gambling question became the language of uncertainty itself.

Stochastic Process

Random Walk

Each step: equally likely up or down. Click to regenerate.

E[X]
E[X] = Σ xi · P(xi)

The long-run average value of repetitions of the same experiment.

H
T
Flip the coin
0H : 0T