Roll a die and write down the number on the top face. This will be a random number because it was generated by a random process whose outcome could not be predicted in advance. Mathematicians, scientists, and even gamblers often want to simulate chance occurrences. Rather than roll dice, toss coins, or pick balls from urns--methods that were common 100 years ago--today they often use random numbers drawn from a table or generated by a computer. Random numbers are used in casino games, state lotteries, computer games, clinical trials of new drugs, simulations of random walks, encryption techniques, modeling molecular behavior, testing computer programs, and even in computer graphics for rendering realistic-looking images. Monte Carlo simulations refer to algorithms that use random numbers and statistical sampling to estimate calculations in physics, probability theory, and other areas of mathematics and science.
In 1927, L. Tippitt of the Biometric Laboratory at the University College in London compiled and published a table of 41,600 numbers that were obtained from the middle digits of the measured areas of English parishes. He meant for these to be used as a source of random numbers. In the 1950s, the RAND Corporation built a special machine to generate pseudorandom binary bits of 0 or 1 that were then used to produce a table of one million random decimal digits. Statistical tests can be performed on tabulations such as these to test how random the numbers really are. For example, in a large sample of random numbers, each of the ten digits from 0 to 9 should appear about one-tenth of the time. More than one test, however, must often be used to guarantee a reasonable degree of randomness.
John Von Neumann was an early pioneer in the development of computer-based random number generators. The use of a computer program provides a problem, however, for how can the numbers be random if they are obtained from a deterministic computer algorithm? Most random number algorithms will produce the same list of values if the algorithm starts with the same initial number, called the seed. Computer generated numbers are therefore called pseudorandom.
The most common method for generating pseudorandom numbers is to use a linear congruential generator invented in 1951 by Derrick H. Lehmer (1905-1991). The method uses three integers a, b, and m. Given an initial seed number x, the generator computes ax+b (mod m), that is, the remainder when ax+b is divided by m. The calculation is then repeated with this number as the next seed to the generator. With an appropriate choice for a, b, and m, each new number should appear to be independent of the previously generated numbers. However, because there are only a finite number of possible remainders when a number is divided by m, a researcher using this method could use up all the pseudorandom numbers before finishing an experiment. In 1991, George Marsaglia and Arif Zaman at Florida State University designed a random number generator capable of producing a sequence of pseudorandom numbers with a period of at least 10250 before the numbers begin to repeat. In the modern equivalent of the random number tables, Marsaglia made freely available a CD-ROM containing almost five billion random bits that can be combined in different ways to give random numbers. The bits were made by combining the output of the Marsaglia-Zaman generators with several different sources of electronic noise, digitized music, and digitized pictures.
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