Probability theory is a part of stochastics in mathematics. It deals with the calculation of probabilities for events in random experiments. It is therefore often difficult to grasp, because random experiments can always end differently than one suspects or than the probability predicts.
The theory of probability probably arose from the desire to be able to influence games of chance. The year 1654 is said to be the year when probability theory began: a French philosopher wrote a letter to his mathematician friend Blaise Pascal in that year, complaining that he was regularly wrong at a game of dice. Then, with the help of the mathematician Pierre de Fermat, Pascal made some calculations that are considered the origin of probability theory.
What will you learn in the Probability Chapter?
Basics of probability theory
In the first section you will learn the basics of probability theory. Here you can find out what the terms random experiment or random test, event, Laplace experiment and probability mean. You will also learn about de Morgan’s laws, the law of large numbers and so-called Venn diagrams. Relative and absolute frequencies are also dealt with. Finally, you will find an article summarizing rules and theorems for calculating with probabilities.
Multi-level random experiments
If you repeat a random experiment several times, you can create a multi-stage random experiment. You can find a detailed explanation of what multi-stage random experiments are in the article multi-stage random experiments – basics. In this section you will also get to know the path rules, which are essential for multi-stage random experiments and make working with tree diagrams a lot easier! You can easily read here what makes the difference between the first and second path rule!
conditional probability
Conditional probability is when, in a random experiment, the occurrence of one event depends on the occurrence of another event. For example, the probability that a randomly chosen person has a shoe size of 35 is significantly higher under the condition that the person is younger than 15 than under the condition that the person is at least 16 years old.
Since the four-field table and the tree diagram are of great importance for conditional probabilities, these two topics are taken up in this chapter. You can also get to know the multiplication theorem or product theorem, what stochastic independence is and what Bayes’ theorem says. You will also find an article about the theorem of total probability and the theorem of addition.
Finally, there is an article about the goat problem, which was based on a TV show and has puzzled many mathematicians for years!
simulation
A simulation is a replica of a random experiment. The urn experiment is a popular and commonly used simulation of random experiments. There are different ways of drawing from the urn: with or without replacement and with or without paying attention to the order. You will learn about these options in the Combinatorics section. In the Simulation chapter, we first concentrate on some basics of simulation and then you get to know the so-called Monte Carlo method, in which a random test is simulated using random numbers.
combinatorics
Combinatorics is a branch of mathematics that is not only used in stochastics, but also in set theory and group theory. Important mathematicians who shaped the field of combinatorics were Pascal, Fermat, Leibniz and Bernoulli.