2
RANDOM NUMBERS
3.1
e) Decision making. There are reports that many executives make their deci-
sions by flipping a coin or by throwing darts, etc. It is also rumored that some
college professors prepare their grades on such a basis. Sometimes it is impor-
tant to make a completely “unbiased” decision; this ability is occasionally useful
in computer algorithms, for example in situations where a fixed decision made
each time would cause the algorithm to run more slowly. Randomness is also an
essential part of optimal strategies in the theory of games.
f) Recreation. Rolling dice, shuffling decks of cards, spinning roulette wheels,
etc., are fascinating pastimes for just about everybody. These traditional uses
of random numbers have suggested the name “Monte Carlo method,” a general
term used to describe any algorithm that employs random numbers.
People who think about this topic almost invariably get into philosophical
discussions about what the word “random” means. In a sense, there is no such
thing as a random number; for example, is 2 a random number? Rather, we speak
of a sequence of independent random numbers with a specified distribution, and
this means loosely that each number was obtained merely by chance, having
nothing to do with other numbers of the sequence, and that each number has a
specified probability of falling in any given range of values.
A uniform distribution on a finite set of numbers is one in which each possible
number is equally probable. A distribution is generally understood to be uniform
unless some other distribution is specifically mentioned.
Each of the ten digits 0 through 9 will occur about & of the time in a
(uniform) sequence of random digits. Each pair of two successive digits should
occur about & of the time, etc. Yet if we take a truly random sequence of a
million digits, it will not always have exactly 100,000 zeros, 100,000 ones, etc. In
fact, chances of this are quite slim; a sequence of such sequences will have this
character on the average.
Any specified sequence of a million digits is equally as probable as the
sequence consisting of a million zeros. Thus, if we are choosing a million digits at
random and if the first 999,999 of them happen to come out to be zero, the chance
that the final digit is zero is still exactly &, in a truly random situation. These
statements seem paradoxical to many people, but there is really no contradiction
involved.
There are several ways to formulate decent abstract definitions of random-
ness, and we will return to this interesting subject in Section 3.5; but for the
moment, let us content ourselves with an intuitive understanding of the concept.
At first, people who needed random numbers in their scientific work would
draw balls out of a “well-stirred urn” or would roll dice or deal out cards. A
table of over 40,000 random digits, “taken at random from census reports,” was
published in 1927 by L. H. C. Tippett. Since then, a number of devices have
been built to generate random numbers mechanically; the first such machine was
used in 1939 by M. G. Kendall and B. Babington-Smith to produce a table of
100,000 random digits, and in 1955 the RAND Corporation published a widely
used table of a million random digits obtained with the help of another special