#### Probability Theory

Summer 2020, Grade: 1.7

Mathematical probability theory is concerned with phenomena that either cannot be properly modelled by deterministic quantities (such as dispersion or financial markets) or are fundamentally random (such as quantum systems).

History: Classical considerations of probability and chance date back to the sixteenth century, but its modern formulation in terms of measure theory wasn’t developed until the 1930s by A. Kolmogorov. The latter approach postulates the existence of a probability space consisting of a sample set, a sigma-algebra of measurable events, and a probability measure, as well as what is called a random variable: a measurable function from the sample space into some measure space of outcomes (often the real numbers). The sample space can be interpreted as all possible states the universe could be in when the random experiment happens. The sigma-algebra indicates for which events it is decidable whether the event happened or not by just observing the random variable, whereas the probability measure gives the probability that a certain state of the universe happens. Finally, the random variable itself represents the random experiment, assigning to each state in the sample space an outcome. The problem here is that the probability space is very inaccessible, which is why, instead of taking the probability space as the central object of study, one considers a probability measure on the target space of the random variable: its distribution. The distribution of a random variable indicates with what probability a certain outcome of an experiment happens. For example, a random variable could be normally distributed.

As opposed to statistics, which aims to make statements about the underlying random process which generates data, probability theory is concerned with making predictions about the data given by the random process. In a sense, they are inverse to one another.

Inner workings: The main goal of PT is to deduce not necessarily concrete probabilities of specific events happening or not, but rather to prove that an event will certainly happen or not. Many probabilistic systems are that way because they depend on many variables, each for themselves is next to impossible to control. However, all taken together show predictable behaviour. For example, whether a fair coin toss returns head or tail is not predictable, but with probability 1 the sample average of more and more coin tosses is 50% head and 50% tails. This is done by limit theorems such as the central limit theorem or the law of large numbers.

Application in Pure Mathematics: Apart from its obvious role in mathematical statistics, PT can be employed in very pure areas of research as well such as the “probabilistic method” in combinatorics. Pioneered by P. Erdös, the existence of an object A can be proven by showing that choosing an object at random returns A with positive probability.

Application in Applied Mathematics: From computer science, chemistry and physics to biology, life sciences and economics, PT always find application when information is random or appears to be random.