= Probability theory
{wiki}
Probability theory describes uncertainty through a <probability space>: possible outcomes form a <sample space>, observable questions are <events>, and a <probability measure> assigns them numbers between zero and one. The <Kolmogorov axioms> make those assignments consistent. They let us combine information about several <events>, update it through <conditional probability>, and study <random variables> without listing every possible outcome separately.
For example, roll two dice whose results are <independent random variables>, each with a <discrete uniform distribution> on $\{1,\ldots,6\}$. There are 36 equally likely ordered outcomes. Only $(1,6),(2,5),\ldots,(6,1)$ have sum seven, so that <event> has <probability> $6/36=1/6$. The sum itself is not uniformly distributed: sum two has only one outcome. Choosing a <probability distribution> for the elementary outcomes is part of the model; equal likelihood does not follow merely from counting the named possibilities.
The subject becomes especially useful when complete outcome lists are enormous or infinite. <Independence> turns a joint calculation into simpler pieces, <expected value> summarizes a <random variable>, and the <law of large numbers> explains how averages can become predictable even when individual observations are uncertain. The <central limit theorem> describes the scale and shape of many remaining fluctuations. These are distinct claims: a small <variance> of an average does not make every observation close to its <expected value>.
There is a satisfying unity here: dice, hidden causes, and <Brownian motion> all use the same rules for <events> and <conditional probability>. A model still needs justification. Its exact deductions describe what its assumptions imply; comparison with observations belongs to <statistical inference>.
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