Infinite monkey theorem

The infinite monkey theorem states that an endless sequence of random characters will almost surely contain any specified finite text, provided each character has a positive probability of occurring.

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Almost-sure convergence: A sequence of random variables converges almost surely when it converges with probability one. The theorem's guarantee is probabilistic certainty in this precise sense.
Probability space: A mathematical model consisting of outcomes, events, and a probability measure. The theorem's random text is defined within a probability space.
Émile Borel: Émile Borel was a French mathematician who helped develop measure theory and probability. His 1913 discussion helped establish the monkey-and-typewriter formulation.
Random search: Random search explores a space of possibilities by sampling candidates without a directed strategy. The theorem is an extreme result about the eventual success of undirected search.
Finite monkey theorem: A finite monkey theorem estimates the probability that random typing produces a specified text within a finite number of keystrokes. Finite runtimes give probabilities, not the infinite theorem's probability-one conclusion.
Borel–Cantelli lemma: A probability theorem connecting the sum of event probabilities to whether infinitely many events occur. It formalizes why repeated independent opportunities eventually yield the text.
Random variable: A function that assigns a numerical value to each outcome in a probability space. Character choices can be modeled as random variables.
Measure theory: Measure theory studies generalized notions of size, including probability and length. The theorem's rigorous probability language grew from measure-theoretic foundations.
Brute-force search: Brute-force search tests possible candidates systematically until it finds one that meets a condition. It contrasts random typing with exhaustive, ordered enumeration of strings.
Law of large numbers: The law of large numbers states that sample averages approach their expected value as sample size grows. It concerns long-run averages, not the appearance of every specified finite pattern.