Applications of randomness
Random drawing of lots has been used throughout recorded history as an unbiased way to distribute risks or rewards. Also common have been games of chance (see page 968). Randomness is in general a convenient way to allow local decisions to be made while maintaining overall averages. In biological organisms it is used in determining sex of offspring, as well as in achieving uniform sampling, say in foraging for food. (Especially in antiquity, all sorts of seemingly random phenomena have been used as a basis for fortune telling.)
The notion of taking random samples as a basis for making unbiased deductions has been common since the early 1900s, notably in polling and market research. And in the past few decades explicit randomization has become common as a way of avoiding bias in cases such as clinical trials of drugs.
In the late 1800s it was noted in recreational mathematics that one could find the value of π by looking at randomly dropped needles. In the early 1900s devices based on randomness were built to illustrate statistics and probability (see page 312), and were used for example to find the form of the Student t-distribution. With the advent of digital computers in the mid-1940s Monte Carlo methods (see page 968) were introduced, initially as a way to approximate processes like neutron diffusion. (Similar ideas had been used in 1901 by Kelvin to study the Boltzmann equation.) Such methods became increasingly popular, especially for simulating systems like telephone networks and particle detectors that have many heterogeneous elements—as well as in statistical physics. In the 1970s they also became widely used for high-dimensional numerical integration, notably for Feynman diagram evaluation in quantum electrodynamics. But eventually it was realized that quasi-Monte Carlo methods based on simple sequences could normally do better than ones based on pure randomness (see page 1085).
A convenient way to tell whether expressions are equal is to evaluate them with random numerical values for variables. (Care must be taken with branch cuts and bounding intervals for inexact numbers.) In the late 1970s it was noted that by evaluating PowerMod[a, n - 1, n] 1 for several random integers a one can with high probability quickly deduce PrimeQ[n]. (In the 1960s it had been noted that one can factor polynomials by filling in random integers for variables and factoring the resulting numbers.) And in the 1980s many such randomized algorithms were invented, but by the mid-1990s it was realized that most did not require any kind of true randomness, and could readily be derandomized and made more predictable. (See page 1085.)
There are all sorts of situations where in the absence of anything better it is good to use randomness. Thus, for example, many exploratory searches in this book were done randomly. And in testing large hardware and software systems random inputs are often used.
Randomness is a common way of avoiding pathological cases and deadlocks. (It requires no communication between components so is convenient in parallel systems.) Examples include message routing in networks, retransmission times after ethernet collisions, partitionings for sorting algorithms, and avoiding getting stuck in minimization procedures like simulated annealing. (See page 347.) As on page 333, it is common for randomness to add robustness—as for example in cellular automaton fluids, or in saccadic eye movements in biology.
In cryptography randomness is used to make messages look typical of all possibilities (see page 598). It is also used in roughly the same way in hashing (see page 622). Such randomness must be repeatable. But for cryptographic keys it should not be. And the same is true when one picks unique IDs, say to keep track of repeat web transactions with a low probability of collisions. Randomness is in effect also used in a similar way in the shotgun method for DNA sequencing, as well as in creating radar pulses that are difficult to forge. (In biological organisms random diversity in faces and voices may perhaps have developed for similar reasons.)
The unpredictability of randomness is often useful, say for animals or military vehicles avoiding predators (see page 1105). Such unpredictability can also be used in simulating human or natural processes, say for computer graphics, videogames, or mock handwriting. Random patterns are often used as a way to hide regularities—as in camouflage, security envelopes, and many forms of texturing and distressing. (See page 1077.)
In the past, randomness was usually viewed as a thing to be avoided. But with the spread of computers and consumer electronics that normally operate predictably, it has become increasingly popular as an option.
Microscopic randomness is implicitly used whenever there is dissipation or friction in a system, and generally it adds robustness to the behavior that occurs in systems.