Probability
Laws and principles filed under Mathematics & Probability / Probability.
Probability
Every guide includes meaning, mechanism, applications, limitations, related concepts, and traceable references.
Fundamental probability theorem
Law of Large Numbers
Sample averages, convergence, weak and strong forms, finite-sample uncertainty, and the gambler fallacy.
Conditional probability theorem
Bayes' Theorem
Evidence updates a prior belief by comparing how expected that evidence is under competing possibilities.
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.
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.
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.
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.