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What Is Randomness?
BBC Radio 4’s ‘In Our Time’ today looked at the issue of inequality. The In Our Time website has a link to the program on iPlayer, in case you missed it the first time around.
What does random mean? Indeed, a random event is not deterministic, that is, it is impossible to determine another result based on previous results, or on something 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 it be so? Because in theory, many processes that we consider random, such as particle shedding, are in fact deterministic. If you knew its exact position, size, etc., you could, theoretically, determine the result of throwing the die.
The ancient Greek philosopher and mathematician Democritus (c. 460 BC – c. 370 BC) was a member of the group known as the Atomists. This ancient group were the pioneers of the concept that all matter can be broken down into its basic building blocks, atoms. Democritus decided that there is no such thing as true coincidence. He gave the example of two men who met at a well, which both saw as pure chance. What they didn’t know is that the meeting might have been pre-arranged by their families. This can be thought of as an analogy for playing the game of determinism: there are factors that determine the outcome, even if we can’t exactly measure or control them.
Epicurus (341 BC – 270 BC), the later Greek mathematician, disagreed. Even though he didn’t know how small they really were, he suggested that they were moving randomly in their tracks. No matter how well we understand the laws of motion, randomness will always be implied by this fundamental property of atoms.
Aristotle worked further on probability, but it remained a non-mathematical pursuit. He divided everything into certain, probable and unknown, for example, he wrote about the consequences of throwing the skull bones, early teeth, as unknown.
Like many other areas of mathematics, the subject of probability and probability did not re-emerge until the Renaissance in Europe. Mathematician and gambler Gerolamo Cardano (September 24, 1501 – September 21, 1576) correctly recorded the probabilities of rolling a six with one tooth, a double six with two teeth, and a triple with three. He was the first to notice, or at least record, that you roll 7s with 2s more than any other number. These statements formed part of his handbook for players. Cardano suffered a lot because of his addiction to gambling (sometimes he took all his family’s belongings, became a poor man and was involved in wars). This book was his way of telling his fellow gamblers how much to bet and how to stay out of trouble.
In the 17th century, Fermat and Pascal collaborated and developed a more formal theory of probabilities and assigned numbers to probabilities. Pascal developed the idea of expected value and famously used a probability argument, Pascal’s Wager, to justify his belief in God and his precious life.
Today there are sophisticated tests that can be performed on a sequence of numbers to determine whether or not the sequence is truly random, or whether it was determined by a formula, human, or some other means. For example, is the number 7 a tenth of a time (plus or minus an allowed error)? The number 1 is followed by another 1 a tenth of a time?
A series of increasingly sophisticated tests can be put into action. We have the “poker test”, which analyzes the numbers in groups of 5 to see if there are two pairs, three of a kind, etc. The Chi Squared test is another statistician’s favorite. Because a particular pattern has occurred, it will give a probability and a level of confidence that it was generated by a random process.
But none of these tests are perfect. There are deterministic sequences that appear random (pass all tests) but are not. For example, the numbers of the odd number π look like a random sequence, and pass all tests of randomness, but of course, they are not. π is a deterministic sequence of numbers – mathematicians can count it to as many decimal places as they want, as long as computers are powerful enough.
Another natural, seemingly random division is that of prime numbers. Riemann’s hypothesis provides a way to calculate the distribution of prime numbers, but it remains incomplete and no one knows if the hypothesis is valid for very large values. However, like the numbers in the irrational number π, the distribution of prime numbers passes all tests of randomness. It remains decisive, but unpredictable.
Another useful measure of randomness is a statistic called Kolmogorov Complexity, named after the 20th century Russian mathematician. Kolmogorov’s complex is the shortest description of a sequence of numbers, for example the sequence 01010101… can simply be called “Repeat 01”. This is a very brief description, which shows that the order is definitely not random.
However, for a real random sequence, it is impossible to describe the sequence of numbers in a simplified way. The description will be as long as the string itself, which means that the string will appear random.
Over the last two centuries, scientists, mathematicians, economists and many others have begun to realize that sequences of random numbers are crucial to their work. And so in the 19th century, methods were developed for random numbers. Dice, but can be biased. Walter Welden and his wife drilled a set of 12 teeth 26,000 times at their kitchen table for months, but the data was found to be incorrect because they were poorly embedded in the teeth, which seems 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 mistakes. One of the most successful methods was that used by John von Neumann, who pioneered the mean square method, in which a 100-digit number is squared, the middle 100 digits are subtracted from the result, and squared again, and so on. Very quickly, this process yields a set of numbers that pass all randomness tests.
In the 1936 United States presidential election, all public opinion polls showed close results, with Republican Party candidate Alf Landon winning. In the event, the result was for the Democratic Party Franklin D Roosevelt. Surveys used poor sampling techniques. In their high-tech efforts, they reached out to people by phone to ask them about their voting intentions. In the 1930s, wealthy people—mostly Republican voters—were more likely to own a telephone, and so poll results were heavily biased. In surveys, accurate sampling of the sample population is of prime importance.
Similarly, it is also very important in medical experiments. Selecting a biased sample group (eg too female, too young, etc.) can make a drug seem less likely to work, bias the trial, and potentially have dangerous consequences.
One thing’s for sure: humans aren’t very good at creating random sequences, and they’re not very good at finding them either. When tested with two patterns of dots, a person is particularly bad at deciding which pattern is randomly generated. Also, when trying to generate a random sequence of numbers, very few people include features such as numbers occurring three times in a row, which is a very important feature of random sequences.
But is anything truly random? Going back to the loop we thought at the beginning, where knowing the exact initial conditions would allow us to predict the outcome, this is certainly true of any physical process that produces a set of numbers.
Well, until now, atomic and quantum physics have come closest to providing us with truly unpredictable events. Currently, it is impossible to determine exactly when a radioactive material will decay. It seems random, but maybe we just don’t understand. Currently, it is probably the only way to generate truly random sequences.
Ernie, the UK Government’s premium contact number producer, is now in its fourth revival. It should be random, to give all premium bond holders in the country an equal chance of winning. It contains a chip that uses thermal noise inside itself, that is, the rate of movement of electrons. Government statisticians perform tests of the number series that this produces, and they actually pass the tests for randomness.
Other applications include: randomizing prime numbers used in internet transactions, encrypting your credit card number. National Lottery machines use a set of very light balls and air currents to mix them up, but like the flowers, this can be predicted in theory.
Finally, the Met Office uses a set of random numbers for the ensemble forecasts. Sometimes the weather is difficult to predict because of the well-known “chaos theory” – that the final state of the atmosphere is highly dependent on the actual initial conditions. It’s impossible to measure the initial conditions with anything like the precision required, so atmospheric scientists run their computer models through different scenarios, with the initial conditions varying slightly in each. This results in a set of different predictions and a weather forecaster that talks about percentage chances, rather than guesswork.
See also: In our time.
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