The BBC Radio 4 program ‘In Our Time’ examined the issue of randomness today. The In Our Time website has a link to the show on the iPlayer, if you missed it the first time.
What is meant by randomness? Well, a truly random event is not deterministic, that is, it is not possible to determine the next result, based on previous results or anything else.
In fact, random processes are very important in many areas of mathematics, science, and life in general, but truly random processes are very difficult to achieve. Why should this be the case? Because, in theory, many processes that we consider random, such as rolling a die, are in fact deterministic. Theoretically, you could determine the outcome of the dice roll if you knew its exact position, size, etc.
The ancient Greek philosopher and mathematician Democritus (ca. 460 BC – ca. 370 BC) was a member of the group known as the atomists. This group of ancients pioneered the concept that all matter can be subdivided into its fundamental building blocks, atoms. Democritus decreed that true randomness did not exist. He gave the example of two men who meet in a well, and both consider that their meeting was pure chance. What they did not know is that the meeting was probably previously arranged by their families. This can be considered an analogy of the deterministic roll of the dice: there are factors that determine the result, even if we cannot measure or control them precisely.
Epicurus (341 BC – 270 BC), a later Greek mathematician, disagreed. Although he had no idea how small the atoms actually were, he suggested that they deviate randomly in his path. No matter how well we understand the laws of motion, there will always be randomness introduced by this underlying property of atoms.
Aristotle worked more on probability, but it remained a non-mathematical quest. He divided all things into certain, probable and unknowable, for example, writing about the result of throwing knuckle bones, early dice, as unknowable.
As with many other areas of mathematics, the subject of randomness and probability did not resurface in Europe until the Renaissance. The mathematician and gambler Gerolamo Cardano (September 24, 1501 – September 21, 1576) correctly recorded the probabilities of rolling a six with one die, a double six with two dice, and a triple with three. He was the first person to notice, or at least record, the fact that you are more likely to roll 7 with 2 dice than with any other number. These disclosures were part of his player manual. Cardano had suffered terribly from his love of gambling (sometimes he pawned all his family’s belongings, ended up in a poor house and in fights). This book was his way of telling his fellow players how much they should bet and how to avoid getting into trouble.
In the 17th century, Fermat and Pascal collaborated and developed a more formalized theory of probability, and probabilities were assigned numbers. Pascal developed the idea of an expected value and used a probabilistic argument, Pascal’s Wager, to justify his belief in God and his virtuous life.
Today there are sophisticated tests that can be performed on a sequence of numbers to determine whether the sequence is truly random or whether it has been determined by a formula, a human, or some other means. For example, does the number 7 occur one-tenth of the time (plus or minus some allowed error)? Is the digit 1 followed by another 1 one-tenth of the time?
An increasingly sophisticated series of tests can be launched. We have the “poker test”, which analyzes numbers in groups of 5, to see if there are two pairs, triplets, etc., and compares the frequency of these patterns with those expected in a truly random distribution. The Chi-square test is the favorite of other statisticians. Since a particular pattern has occurred, it will give a probability and a confidence level that it was generated by a random process.
But none of these tests is perfect. There are deterministic sequences that seem random (they pass all tests) but they are not. For example, the digits of the irrational number π look like a random sequence and pass all tests of randomness, but of course it is not. π is a deterministic sequence of numbers: mathematicians can calculate it to as many decimal places as they want, with powerful enough computers.
Another seemingly random distribution that occurs naturally is that of prime numbers. The Riemann Hypothesis provides a way to calculate the distribution of prime numbers, but it remains unsolved and no one knows whether the hypothesis is still valid for very large values. However, like the digits of the irrational number π, the distribution of prime numbers passes all tests for randomness. It is still deterministic, but unpredictable.
Another useful measure of randomness is a statistic called Kolmogorov Complexity, named after the 20th century Russian mathematician. Kolmogorov complexity is the shortest possible description of a sequence of numbers, for example the sequence 01010101 …. could be simply described as “Repeat 01”. This is a very short description, indicating that the sequence is certainly not random.
However, for a truly random sequence, it would be impossible to describe the sequence of digits in simplified form. The description would be as long as the sequence itself, indicating that the sequence would appear to be random.
During the last two centuries, scientists, mathematicians, economists, and many others have begun to realize that random number sequences are very important to their work. And so, in the 19th century, methods were devised to generate random numbers. Says, but may be biased. Walter Welden and his wife spent months at their kitchen table rolling a game of 12 dice more than 26,000 times, but this data was found to be flawed due to dice biases, which seems like a terrible shame.
The first published collection of random numbers appears in a 1927 book by Leonard HC Tippet. After that, there were many attempts, many failed. One of the most successful methods was the one used by John von Neumann, who pioneered the mean square method, in which a 100-digit number is squared, the middle 100 digits are taken out of the result, and rewritten. square, and so on. Very quickly, this process produces a set of digits that pass all randomness tests.
In the 1936 American presidential election, all opinion polls pointed to a tight result, with a possible victory for Republican Party candidate Alf Landon. In the event, the result was a landslide victory for Franklin D. Roosevelt of the Democratic Party. The opinion pollsters had chosen poor sampling techniques. In their attempts to be high-tech, they had phoned people to ask about their voting intentions. In the 1930s, wealthier people, mostly Republican voters, were much more likely to own a phone, so the poll results were deeply skewed. In surveys, the true randomization of the sample population is of primary importance.
Likewise, it is also very important in medical tests. Choosing a biased sample set (eg, too many women, too young, etc.) can make a drug seem more or less likely to work, skewing the experiment, with possibly dangerous consequences.
One thing is certain: humans are not very good at producing random sequences and they are not very good at detecting them either. When testing with two dot patterns, a human being is particularly bad at deciding which pattern has been randomly generated. Similarly, when trying to create a random sequence of numbers, very few people include features like digits that appear three times in a row, which is a very prominent feature of random sequences.
But is there anything truly random? Going back to the dice we considered at the beginning, where knowledge of the precise initial conditions would have allowed us to predict the outcome, surely this is true for any physical process that creates a set of numbers.
Well, so far atomic and quantum physics have come closer to providing us with truly unpredictable events. To date, it is impossible to determine precisely when a radioactive material will decompose. It seems random, but maybe we just don’t get it. At the moment, it is still probably the only way to generate truly random sequences.
Ernie, the UK government’s premium bond number generator, is now in his fourth reincarnation. It must be random, so that all premium bondholders in the country have an equal chance of winning a prize. It contains a chip that takes advantage of thermal noise within itself, that is, the momentum of electrons. Government statisticians test the number sequences that this generates and, in fact, pass randomness tests.
Other applications are: the random prime numbers used in Internet transactions, encrypting your credit card number. The machines of the National Lottery use a set of very light balls and currents of air to mix them, but like dice, this could, in theory, be predicted.
Finally, the Met Office uses sets of random numbers for its established forecasts. It is sometimes difficult to predict the weather due to the well-known “chaos theory”: that the final state of the atmosphere is highly dependent on the precise initial conditions. It is impossible to measure initial conditions with the required precision, so atmospheric scientists feed their computer models with several different scenarios, with initial conditions varying slightly for each. This results in a set of different forecasts and a weather presenter speaking in percentage of probabilities, rather than certainties.
See also: In our time.