randomized

Randomized refers to methods and processes that deliberately incorporate randomness, typically via random sampling or random numbers. The goal of randomization is to prevent bias, enable efficient sampling, or achieve better average-case performance in procedures that are otherwise deterministic.

In statistics and experimental design, randomized methods are central. A randomized controlled trial assigns participants to treatment and control groups by chance, using predefined rules such as simple randomization or block randomization. Randomization helps balance known and unknown confounding factors across groups, supporting valid inference about treatment effects. In survey sampling, random sampling methods select units randomly to obtain representative estimates with quantifiable margins of error.

In computer science and mathematics, randomized algorithms use randomness as an auxiliary resource. They are typically categorized as Monte Carlo algorithms, which produce results that may be incorrect with small probability but have bounded error, and Las Vegas algorithms, which always produce correct results but have a random running time. Randomized methods are used in numerical integration, optimization, primality testing, and graph algorithms, among others. Well-known examples include randomized quicksort, Monte Carlo integration, and randomized selection algorithms. Randomized methods also underpin various probabilistic data structures and techniques, such as skip lists, treaps, and randomized hashing, which rely on randomness to maintain expected performance guarantees.

In practice, the quality and source of randomness matter: true randomness or high-quality pseudorandom number generators are used, with cryptographic applications requiring strong unpredictability. Overall, randomized approaches offer a flexible toolset for dealing with uncertainty, bias, and complexity across disciplines.