Assume you’re on a game show and you are asked to pick one out of three doors and win whatever is behind it. You know that behind one door is a car; behind the others, goats. Let’s say you pick door Number 1, and the host, who knows what’s behind the other doors, opens another door, say Number 3, revealing a goat and asks you if you wish to change your selection to door No. 2 and win its contents. You are now facing a dilemma: “Is it to your advantage to take the switch?”
Mathematics, more specifically, probability theory suggests that the optimal strategy is switching, doubling your initial probability of winning from ⅓ to ⅔. This may seem counter-intuitive, however, supposing that your goal is to find in your first choice a door containing a goat (something that has a probability of ⅔) and always switching afterwards, means that in the end you will end up with a car two out of three times. It should be noted that, from the host’s viewpoint there is no uncertainty whatsoever, but for the player, who does not know which door contains the prize, his ignorance transforms into uncertainty.
From the ancient times, it is known that people were familiar with random experiments; a fancy alternative phrase for gambling. Indeed, mythology is filled with stories in which the outcome of a game of chance determined the course of events. For instance, by a throw of dice, Zeus, Poseidon and Hades split the universe. More importantly, in Egyptian mythology, Thoth, the ancient Egyptian god of science, gambled with the Moon and won some moonlight in a game of dice, proving that it is not the lucky one that wins, but the one who has developed a better understanding of the odds of winning in various situations than any of his opponents. Other examples throughout history include Emperor Marcus Aurelius being regularly accompanied by his personal croupier and Roman soldiers throwing dice for Christ’s clothes.
Although it would be centuries later that a mathematical theory of probability would start to form, it would be naive to think that no quantitative or numerical concept of probability was used before. For example, in the capital of the Byzantine Empire, the chariot races of the Hippodrome of Constantinople were hugely popular with the masses and betting was not uncommon for many spectators. Surely, at least one way of calculating the odds for these occasions was available to the organisers, otherwise they would be either unable to pay out the winnings if the odds were more in favour of the risk-takers or no one would bet if the odds were hugely rigged against them. Another instance of not only controlling randomness, but also forecasting and inventing the very first usage of what we now call a derivate in finance, is described in Aristotle’s Politics. The story places Thales of Miletus, one of the Seven Sages of Greece, in wintertime, using his knowledge of astronomy (and not astrology) that there was going to be a large crop of olives. Thus, he hired all the olive-presses in Miletus and Chios at a low rent; and when the harvest season arrived, there was a sudden increase in demand for a number of presses at the same time, and by subletting them out on what terms he liked, he gained a large sum of money, proving that it is easy for philosophers to be rich if they like, but this is not what they care about.
Surprisingly, it is only after the Black Death pandemic, peaking in Europe in mid-14th century that the first attempts for a rigorous scientific exploration of randomness are initiated. Filippo Calandri, a century later, provides one of the first analysis of a game of chance that is stopped prematurely, and the prize pot must then be fairly split among the players (a similar problem is solved in an anonymous Italian mathematical manuscript of about 1400). Later, scientists such as Huygens and Leibniz, would apologise in their manuscripts for studying trivial subjects like dice problems, as their use was to practice for more important matters. Indeed, “standing on the shoulders of giants” and using statistics, Gauss was able to predict the orbit of celestial objects, and John Snow, one of the founders of modern epidemiology, traced the source of a cholera outbreak in Soho, London. Lastly, once again after the tragedy of World War I, Kolmogorov, a Soviet mathematician, laid the foundations for modern probability theory and, during World War II, protected Moscow from German bombers by applying statistics and developing a plan of distributing barrage balloons in an optimal way.
From all the stories described above, the following two conclusions can be naturally derived. Firstly, after an event that shakes the world, major advances in various fields follow. Indeed, Renaissance marked the transition from the Middle Ages after the crisis of the Great Bubonic Plague – a period that gave birth to the term quarantine. Secondly, the ones equipped with more knowledge, like Thales, are in a better position to predict future outcomes. Thankfully, in our day, we have many statistical tests available to deal with many uncertain matters, providing everyone with extra knowledge for decision making. For example, pregnancy tests are useful since an approximation of the true situation is given. However, you may not be pregnant, but still, the test says “yes” – this is called a FALSE POSITIVE – or you may be really pregnant, but the test says “no” – this is called a FALSE NEGATIVE. That is why it is recommended to use more than one test or seeing your doctor to put any uncertainty at ease.
The situation described in the example above may seem funny, however, matters get serious when similar outcomes happen when testing for the current Covid-19 virus. In the case of a FALSE NEGATIVE, a person with the virus will go home untreated and although it may not necessarily result in a fatality, the same cannot be said for someone in a high-risk group. The case of a FALSE POSITIVE is the worst scenario for the healthcare system. In this case, healthy individuals will be temporarily occupying limited resources, useful to patients with the virus, and may even result to an overload of the system; setting aside the psychological burden that the individual, family and friends would need to deal with. Thus, the reliability, i.e., the accuracy, of the tests must be extremely high to enhance our decision-making process in this fight against the novel coronavirus.
- Antonios Meimaris is a PhD graduate & Teaching Associate in the Department of Econometrics & Business Statistics at Monash University, Australia and a member of the Stochastic Engineering Dynamics Lab at Columbia University, USA. Before starting his current program, he was a PhD candidate in the Department of Mathematical Sciences at the University of Liverpool and, as a member of the EPSRC and ESRC Centre for Doctoral Training (CDT) in Risk & Uncertainty, I held a Postgraduate Research Studentship (2015 – 2019). He received his first degree in Mathematics from the University of Athens.