BBC Radio 4’s “In Our Time” program examined the topic of randomness in today’s world. If you have missed the program the first time, the In Our Time website provides a link to it on the iPlayer.
What does randomness mean? A truly random event is not predictable. It is impossible to predict the outcome of a random event based on previous outcomes or any other factors.
Although random processes are important in many areas of science, maths, and life, they are extremely difficult to achieve. This is why it should be. Many processes we think of as random, like rolling a die, can actually be considered deterministic. If you had the exact size and position of the dice, you could theoretically predict the outcome.
Democritus, an ancient Greek mathematician, and philosopher was born ca. 460 BC – ca. 370 BC) and was a member of the group called Atomists. These ancients were pioneers in the idea that all matter could be broken down into its basic building blocks, the atoms. Democritus declared that there is no true randomness. Democritus gave the example of two men who met at a well and both believed their meeting was pure chance. They didn’t know that their families had probably planned the meeting. This analogy can be used to explain the deterministic dice roll. Even though we don’t have the ability to control or measure them, there are factors that determine the outcome.
Epicurus (341 BC – 270 BC), a later Greek mathematician, disagreed. He suggested that atoms would drift randomly, even though he didn’t know how small they were. This underlying property of atoms will cause randomness regardless of how well we know the laws of motion.
Aristotle continued to work on probability, but it remained non-mathematical. He separated all things into certain and probable, such as writing about the outcome when throwing knuckle bones.
Like many other areas in mathematics, the topic of randomness and probability didn’t resurface until the Renaissance. Gerolamo Cardano, a mathematician, and gambler was born on 24 September 1501 and died on 21 September 1576. He correctly calculated the probability of throwing a six with one, a double six using two dice, and a three with three dice. He was the first to note, or at the very least record, that 7 is more likely to be thrown with 2 dice than any other number. These revelations were part of his guidebook for gamblers. Cardano’s gambling addiction had caused him severe pain. He had pawned his entire family’s possessions and ended up in a poor home, as well as in fights. He wrote this book to tell fellow gamblers how much to bet and how they can stay out of trouble.
Fermat and Pascal developed a formalized theory of probability in the 17th century. Numbers were then assigned probabilities. Pascal invented the concept of an expected value. He also famously used Pascal’s Wager to support his belief in God, his virtue, and his belief that God is real.
Modern tests can be used to test a sequence of numbers for accuracy. They can determine if the sequence is random or if it was determined by formulas, humans, or other methods. Is the number 7 found one-tenth of all times (plus or minus an allowable error) for example? Is digit 1 followed up by another 1 one-tenth?
A series of increasingly complex tests can be put into practice. The “poker test” analyzes numbers in groups of 5 to determine if there are two, three, or all of the same. It then compares the frequency with the expected patterns in a random distribution. Another favorite statistician is the Chi-Squared test. If a certain pattern has been observed, it will indicate a probability and a level of confidence that it was created by random processes.
None of these tests is perfect. Some deterministic sequences look random and pass all of the tests, but they are not. The digits of p, for example, look random and pass all tests for randomness. If you have powerful computers, mathematicians will be able to calculate p to any number of decimal places.
The prime numbers are another naturally occurring distribution that appears to be random. Although the Riemann Hypothesis allows you to calculate the distributions of primes, it is not yet solved and no one knows if the hypothesis will still be valid for large values. The distribution of primes passes all tests of randomness, just like the digits of the irrational numbers p. It remains deterministic but unpredictable.
A statistic called the Kolmogorov Complexity is another useful measure of randomness. It was named after the Russian mathematician of the 20th century. The Kolmogorov Complexity describes a sequence of numbers in the shortest possible way. For example, the sequence 01010101 …. can be simply described as “Repeat 01”. This description is very brief and indicates that the sequence is not random.
It would be difficult to describe a sequence of digits that is truly random in a simplified way. It would take as much time to describe the sequence as it does the sequence, which would indicate that it would appear random.
Scientists, economists, mathematicians, and others have realized that random numbers are essential to their work over the past two centuries. Methods were developed to generate random numbers in the 19th Century. However, dice can be biased. Walter Welden and his spouse spent months at the kitchen table, rolling 12 dice more than 26000 times. However, these data were flawed due to biases in the dice. This is a shame.
Leonard HC Tippet published the first collection of random numbers in a 1927 book. There were many other attempts to publish random numbers, some of which were unsuccessful. John von Neumann was the first to use the middle-square method. This involves a 100-digit number being squared and the middle 100 digits being extracted from the result. Then, it is squared again and so forth. This process produces digits that can pass all tests of randomness very quickly.
All opinion polls for the 1936 US presidential election indicated a close outcome, with Alf Landon, the Republican Party’s candidate, possibly winning. The result was a landslide victory for Franklin D Roosevelt, a Democratic Party candidate. Bad sampling methods were used by the opinion pollsters. They had called people up to get their vote intentions in an attempt to be high-tech. It was much more common for wealthy people, largely Republican voters, to own a telephone in the 1930s. The results of surveys were therefore biased. It is crucial to randomize the sample population in surveys.
It is equally important for medical tests. A biased sample set (e.g. Too many women, too few young people, etc. A drug can appear less or more likely to work. This could lead to dangerous side effects.
One thing is certain, humans aren’t very good at creating random sequences. They also aren’t very good at spotting them. A test with two patterns of dots showed that a human being can’t distinguish between the generated random pattern and the one created at random. The same goes for creating a random sequence. Very few people can include features like digits appearing three times in one row. This is a very important feature of random sequences.
Is there really anything random? If we look back at the initial conditions of the dice, which allowed us to predict the outcome, then surely the same holds true for any physical process that creates a set number.
Quantum and atomic physics are the closest to providing truly unpredictable events. It’s impossible to predict exactly when radioactive materials will begin to decay. Although it seems random, maybe we don’t know enough. It is the only way to generate random sequences.
Ernie, the UK Government’s premium bond number generator is currently in its fourth incarnation. To give everyone a chance at a prize, it must be random. It has a chip that exploits thermal noise in itself, i.e. The amount of electron movement. The number sequences generated by this test are performed by government statisticians, who pass the tests for randomness.
Other applications include the random prime numbers that are used in internet transactions and encrypting credit card numbers. Although the National Lottery machines use very light balls and air currents to mix them, it could theoretically be predicted.
The Met Office also uses random numbers to generate its ensemble forecasts. It is sometimes difficult to predict whether due to the “chaos theory”, which states that the final state and temperature of the atmosphere are highly dependent upon the initial conditions. Atmosphere scientists use computer models to simulate different scenarios. The initial conditions can be difficult to measure so they feed them different scenarios with slight variations. This creates a variety of forecasts and a weather presenter that speaks in percentages rather than certainties.
