Probability theory is a mathematics branch that mainly deals with the analysis of random phenomena. According to probability theory, the outcome of a random event cannot be determined before it happens, but it can be one of several possible results. In this post, we will take a closer look at this theory and its applications.
What Is Probability Theory?
Probability theory is a mathematics branch that mainly revolves around the chance of an action taking place. There are several probability interpretations, and the theory treats this concept in a rigorous mathematical manner by illustrating it through a set of axioms. These sets of axioms view probability in terms of a probability space, which has values between 0 and 1 that are also called probability measures.
Probability Theory Topics
The topics covered in Probability theory include stochastic processes, probability distributions, and continuous and discrete random variables. The probability theory via stochastic processes helps provide mathematical abstractions of uncertain or non-deterministic processes as well as measured quantities that may be single events or evolve in a random fashion over time.
Although it is impossible to predict random occurrences accurately, probability theory helps describe their behavior through the two main laws of the central limit and large numbers theorem.
Since probability theory serves as the primary mathematical basis for statistics, it is very important to numerous real-life applications that entail quantitative analysis of data. For example, tossing a coin has two possible outcomes; “heads” or “tails.”
Also, tossing two dice has a set of 36 possible outcomes, each of which can assume the values of 1, 2, 3, 4, 5, or 6. This set of all possible outcomes expected in any of these experiments is referred to as a sample space.
The Law of Large Numbers
It is a common suggestion that if a fair coin is tossed several times, then approximately half the time, it will present tails and the other half heads. Additionally, intuition suggests that the more the coin is tossed, the more possible it is that the ratio of the number of tails to the number of heads will be equal. Modern probability theory, also called the law of large numbers, provides a formal representation of this intuitive suggestion.
This law is not usually based on the foundations of probability theory but instead uses these foundations as its theorem. The law of large numbers is often regarded as a pillar in the history of statistical theory as it seeks to link theoretically-derived probabilities with their initial occurrence frequency in the real world.
The mathematical techniques used in probability theory are also applied in the description of complex model systems. These systems are presented with incomplete knowledge of their nature as in sequential estimation or statistical mechanics. Overall, probability theory has many applications and is very common in daily life.