A law earns its place by clarifying how a pattern is produced, not only by naming it.
DOMAIN 0514 PUBLISHED GUIDES
Mathematics & Probability
Formal laws and statistical regularities involving uncertainty, distributions, games, and quantitative structure.
CENTRAL QUESTION
Which patterns follow from structure and uncertainty?
- Working scale
- Axioms to real datasets
- Primary lens
- Proof + statistical inference
14Published
04Defined subfields
FULLSource standard
ENEditorial language
SUBCATEGORY MAP
The field at a glance.
Each subfield has a distinct object of study. Published areas are active now; planned areas show how the collection will grow.
01
Probability
PublishedEvents, uncertainty, conditioning, and long-run behavior.
02
Statistics
PublishedInference, estimation, testing, and uncertainty in data.
03
Distributions
PublishedRecurring shapes in how quantities and frequencies are spread.
04
Game Theory
PublishedStrategic choices made in response to other decision-makers.
PUBLISHED GUIDES
In Mathematics & Probability
Every guide includes meaning, mechanism, applications, limitations, related concepts, and traceable references.
Distributions004
Statistical regularity
Benford's Law
In many naturally occurring datasets, the leading digit is 1 far more often than 9, following a logarithmic distribution.
Probability014
Fundamental probability theorem
Law of Large Numbers
Sample averages, convergence, weak and strong forms, finite-sample uncertainty, and the gambler fallacy.
Probability018
Conditional probability theorem
Bayes' Theorem
Evidence updates a prior belief by comparing how expected that evidence is under competing possibilities.
Statistics019
Asymptotic statistical theorem
Central Limit Theorem
Under broad conditions, properly standardized sums or averages of many independent contributions approach a normal distribution.
Game Theory030
Strategic equilibrium concept
Nash Equilibrium
A strategy profile is a Nash equilibrium when no player can improve their expected payoff by changing strategy alone while the others keep theirs fixed.
Distributions041
Scale-free functional relation and tail model
Power Law
A quantity follows a power law when relative changes in scale produce proportional changes governed by a fixed exponent, often creating heavy-tailed variation.
Game Theory045
Social-choice impossibility theorem
Arrow's Impossibility Theorem
With at least three alternatives, no rank-order aggregation rule can satisfy unrestricted domain, Pareto efficiency, independence of irrelevant alternatives, and non-dictatorship simultaneously.
Probability051
Collision-probability result
Birthday Paradox
A group needs surprisingly few people before the probability that at least two share a birthday becomes substantial, because the number of possible pairs grows quadratically.
Probability060
Distribution-free probability bound
Markov's Inequality
For any nonnegative random variable, the probability of exceeding a positive threshold cannot be greater than the mean divided by that threshold.
Probability087
Probability inequality
Chebyshev's Inequality
For any distribution with finite variance, the probability of being at least k standard deviations from the mean is no greater than 1/k squared.
Statistics088
Statistical selection effect
Regression to the Mean
When two measurements are imperfectly correlated, units selected for an extreme first value tend to be less extreme on a subsequent measurement.
Statistics089
Aggregation reversal
Simpson's Paradox
A trend that appears within every relevant group can weaken, disappear, or reverse after the groups are combined with different weights.
Probability090
Conditional-probability problem
Monty Hall Problem
With three doors, a prize placed uniformly, and a host who always reveals a goat behind an unchosen door, switching wins with probability two thirds.
Game Theory091
Collective-decision theorem
Condorcet Jury Theorem
If voters independently choose between two alternatives and each is more likely than not to be correct, majority accuracy approaches one as the group grows.
CROSS-DOMAIN LINKS
No field stands alone.
These connections show where the domain borrows methods, mechanisms, or evidence from the rest of the library.
Scientific status, operating range, evidence quality, and common misuse stay visible.
Original publications, official records, and serious scholarship take priority.