# Random

### digits chance sequence randomly

The word "random" is used in **mathematics** much as it is in ordinary speech. A random number is one whose choice from a set of numbers is purely a matter of chance; a random walk is a sequence of steps whose direction after each step is a matter of chance; a random **variable** (in **statistics**) is one whose size depends on events which take place as a matter of chance.

Random numbers and other random entities play an important role in everyday life. People who frequent gambling casinos are relieved of their money by slot machines, dice games, roulette, blackjack games, and other forms of gambling in which the winner is determined by the fall of a card, by a ball landing in a wheel' s numbered slot, and so on. Part of what makes gambling attractive is the randomness of the outcomes, outcomes which are usually beyond the control of the house or the player.

Children playing tag determine who is "it" by guessing which fist conceals the rock. Who does the dishes is determined by the toss of a coin.

Medical researchers use random numbers to decide which subjects are to receive an experimental treatment and which are to receive a **placebo**. Quality control engineers test products at random as they come off the line. Demographers base conclusions about a whole population on the basis of a randomly chosen **sample**. Mathematicians use Monte Carlo methods, based on random samples, to solve problems which are too difficult to solve by ordinary means.

For absolutely unbreakable ciphers, cryptographers use pages of random numbers called one-time pads.

Because numbers are easy to handle, many randomizations are effected by means of random numbers. Video poker machines "deal" the cards by using randomly selected numbers from the set 1, 2,..., 52, where each number stands for a particular card in the deck. Computer simulations of traffic patterns use random numbers to mark the arrival or non-arrival of an **automobile** at an intersection.

A familiar use of random numbers is to be seen in the lotteries which many states run. In Delaware's "Play 3" lottery, for example, the winning three-digit number is determined by three randomly-selected numbered balls. The machine that selects them is designed so that the operator cannot favor one ball over another, and the balls themselves, being nearly identical in size and weight, are equally likely to be near the release mechanism when it is activated.

Random numbers can be obtained in a variety of ways. They can be generated by physical means such as tossing coins, rolling dice, spinning roulette wheels, or releasing balls from a lottery machine. Such devices must be designed, manufactured, and used with great care however. An unbalanced coin can favor one side; dice which are rolled rather than tumbled can favor the faces on which they roll; and so on. Furthermore, mathematicians have shown that many **sequences** that appear random are not.

One notorious case of faulty randomization occurred during the draft lottery of 1969. The numbers which were to indicate the order in which men would be drafted were written on slips and enclosed in capsules. These capsules were then mixed and drawn in sequence. They were not well mixed, however, and, as a consequence, the order in which men were drafted was scandalously lacking in randomness.

An interesting source of random numbers is the last three digits of the "handle" at a particular track on a particular day. The handle, which is the total amount bet that day, is likely to be a very large number, perhaps close to a million dollars. It is made up of thousands of individual bets in varying amounts. The first three digits of the handle are anything but random, but the last three digits, vary from 000 to 999 by almost pure chance. They therefore make a well-publicized, unbiased source of winning numbers for both those running and those playing illegal "numbers" games.

Cards are very poor generators of random numbers. They can be bent, trimmed, and marked. They can be dealt out of sequence. They can be poorly shuffled. Even when well shuffled, their arrangement is far from random. In fact, if a 52-card deck is given eight perfect shuffles, it will be returned to its original order.

Even a good physical means of generating random numbers has severe limitations, possibly in terms of cost, and certainly in terms of speed. A researcher who needs thousands of randomly generated numbers would find it impractical to depend on a mechanical means of generating them.

One alternative is to turn to a table of random digits which can be found in books on statistics and elsewhere. To use such tables, one starts from some randomly chosen point in the table and reads the digits as they come. If, for example, one wanted random numbers in the range 1 to 52, and found 22693 35089... in the table, the numbers would be 22, 69, 33, 50, 89,... The numbers 69 and 89 are out of the desired range and would be discarded.

Another alternative is to use a **calculator** or a computer. Even an inexpensive calculator will sometimes have a key for calling up random numbers. **Computer languages** such as Pascal and BASIC include random number generators among the available functions.

The danger in using computer generated random numbers is that such numbers are not genuinely random. They are based on an **algorithm** that generates numbers in a very erratic sequence, but by computation, not chance.

For most purposes this does not matter. Slot machines, for example, succeed in making money for their owners in spite of any subtle bias or regularity they may show. There are times, however where computer-generated "random" numbers are really not random enough.

Mathematicians have devised many tests for randomness. One is to count the **frequency** with which the individual digits occur, then the frequency with which pairs, triples, and other combinations occur. If the list is long enough the "law of large numbers," says that each digit should occur with roughly the same frequency. So should each pair, each triple, each quadruple, and so on. Often, lists of numbers expected to be random fail such tests.

One interesting list of numbers tested for randomness is the digits in the decimal **approximation** for **pi**, which has been computed to more than two and a quarter billion places. The digits are not random in the sense that they occur by chance, but they are in the sense that they pass the tests of randomness. In fact, the decimal approximation for pi has been described as the "most nearly perfect random sequence of digits ever discovered." A failure to appreciate the true meaning of "random" can have significant consequences. This is particularly true for people who gamble. The gambler who plays hunches, who believes that past outcomes can influence forthcoming ones, who thinks that inanimate machines can distinguish a "lucky" person from an "unlucky" one is in danger of being quickly parted from his money. Gambling casinos win billions of dollars every year from people who have faith that the next number in a random sequence can somehow be predicted. If the sequence is truly random, it cannot.

## Resources

### Books

Gardner, Martin. *Mathematical Circus.* New York: Alfred A. Knopf, 1979.

Packel, Edward. *The Mathematics of Games and Gambling.* Washington, DC: The Mathematical Association of America, 1981.

### Periodicals

Stein, S.K. "Existence out of Chaos." *Mathematical Plums.* Ed. Ross Honsberger. Washington, DC: The Mathematical Association of America, 1979.

J. Paul Moulton

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