Probabilities always depend on the events around us

Mathematician Prof. Dr. Barbara Rüdiger works on the calculation of coincidences for the economy and society

"Robin Hood can always hit 100 percent with his bow and arrow, but the rest of us can't," says Barbara Rüdiger, a stochastician in the Faculty of Mathematics and Natural Sciences at Bergische Universität. This is primarily an advantage of such fantasy figures, which are not subject to any influences or imponderables, says the scientist, who studies random calculation. "A coincidence is first of all an event, which can take place, but doesn't have to," she explains. "To calculate it, I look at all the events I am currently observing, then measure the events that are favorable for the coincidence and divide them by the measure of all events." Rüdiger explains it using the example of the cube. "If I roll a non-rigged die, then the probability of an even number appearing is 3:6 because there are three out of six favorable possibilities." But not everything can be counted, she continues, referring to the possibilities an archer has, for example, when he points his arrow at a target. In the case of the Avenger of the Disinherited, you can always assume a 100 percent hit rate, she says, "if I tried to aim, I would calculate the area of the target, by the area of the wall, because my aim would be pretty arbitrary." It always depends on the qualities of the shooter, she said, because if that shooter is skilled, you can describe it very well using a probability distribution called the Gaussian curve, which has a shape of a bell. "The better this shooter is, the narrower the bell curve, the worse he is, the wider the bell curve."

Probabilities exist always and everywhere

Probability theory, or probabilistics, is a branch of stochastics and has applications really everywhere. "Anything that is deterministic, that is, future events that are uniquely determined by preconditions, we can calculate quite accurately without chance using calculus. That depends on conservation laws or conservation properties, but they're always there locally and for a short time." Everything we observe, the scientist formulates, always depends on the environment and if this changes, the events also change. She says this can be seen very well in the example of driving a car. "If I'm steering a car and I want to drive a constant 100 kilometers per hour, I can do that until a car in front or behind me forces me to slow down or speed up. It doesn't just depend on the conservation rate I set for myself. The whole environment will always act to force us to make sudden changes. And these changes are very diverse because we are surrounded by many phenomena. So chance always plays a role."

If you want to calculate probabilities, you have to plan for chance

"Every engineer, all banks and insurance companies use chance," says Ruediger, and it started in the economics field about 100 years ago by Louis Bachelier, the founder of financial mathematics. "He was the first to understand, to describe a security, I have to use chance." And he does this by describing a security using what is known as Brown` s motion. This, in turn, was discovered by the Scottish botanist Robert Brown in 1827, when, under a microscope, he saw irregular and jerky thermal movements of small particles in liquids and gases, which thus changed direction. Physicist Ludwig Boltzmann was then the one a few years later who described the motion of molecules as random. "Boltzmann also understood entropy as a concept for a measure of this disorder microscopically, because this concept is based on chaos." Any modern engineer studying thermodynamics, he said, must take into account this entropy, which is the measure of all ignorance. Rüdiger himself has maintained a regular exchange with a research team at Debeka Insurance for years, where random processes are studied time and again.

Like the probability of winning the lottery: many, many zeros after the decimal point

Chance has often helped one or the other to unexpected wealth. Can mathematical calculations perhaps even influence luck? "I'll say that bachelor's students who have heard our introduction to stochastics should be able to know what the odds of winning the lottery are," the Italian native says with a laugh. Although the chance of winning can be calculated by the fixed number of 49 balls with a probability that is not zero either, but has a zero in front and many, many zeros behind the decimal point.
In the case of roulette, the situation is different again, because with the ball a dynamic comes into play that depends on the throw of the respective croupier and cannot be calculated in many cases.

Does the pandemic also change the probabilities?

"Every crisis brings about a change in the models," says Rüdiger, adding that it is still too early to determine the consequences of the pandemic now. But the 2007 financial crisis, for example, has already provided insights that can be applied to other crises. With regard to securities, he said, it was possible to determine "that they suddenly start making jumps within the crisis, both upward and downward. That is, the whole modeling then has to be changed," she explains. A crisis also changes almost everything in the risk calculation of banks, insurance companies and real estate companies, which often also existed in dependency, she said. "So the real estate market collapses and the risk of that happening implies the collapse of the insurance companies and therefore the banks. Since 2007, you don't study these risks alone, but by modeling them together in the interdependencies." New models are being developed for this purpose, such as the so-called delay equations, which are very complicated because they make the random process dependent not only on the future, but also on the past.

The biggest diamond in the world

The stock market also works with coincidences; investors need to be more risk-averse, but also more fully informed. "You have to have a lot of information there that doesn't just depend on mathematics," the scientist knows and is critical of this financial sector. She wants to keep chance to a minimum in this area and explains her skepticism with an example: "Let's say I have the biggest diamond in the world. Now say I formally divide it into a hundred pieces and am willing to sell forty pieces of it to various other people at a good price. However, the diamond loses value when I physically divide it. Therefore, it remains in my safe. And now I ask, who is willing to buy a part of it, because if one of the other buyers gets into a bind and sells below value, the other share holders have a problem!"
Investments in dependencies behave similarly to probabilities to events, the chance to be calculated can always swing in both positive and negative directions. We will probably never reach the fantastic 100 percent of a Robin Hood.

Uwe Blass (conversation from 12.07.2021)

Prof. Dr. Barbara Rüdiger studied mathematics at the Universities of La Sapienza and Tor Vergata in Rome. In Germany, she first worked under Prof. Dr. Sergio Albeverio in Bochum and Bonn with various fellowships. She then obtained a junior professorship at the University of Koblenz-Landau and from there applied for a full professorship in Wuppertal. Here she has headed the Stochastics Department in the Faculty of Mathematics and Natural Sciences since 2009.