probability theory |# IIT JEE | JEE ADVANCED| MATH HARDEST QUESTION #jee #matholympiad

Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. [2]

The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century (for example the "problem of points").[3] Christiaan Huygens published a book on the subject in 1657.[4] In the 19th century, what is considered the classical definition of probability was completed by Pierre Laplace.[5]

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory.

This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space, introduced by Richard von Mises, and measure theory and presented his axiom system for probability theory in 1933. This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.[6]

Probability is a very popular topic of the Mathematics syllabus of the IIT JEE. It is not considered to be a very simple topic, but it is important to master the topic in order to remain competitive in IIT JEE. Majority of the students find it difficult to master this topic because their basics are not clear. In this chapter, we shall study the methodologies of computing the probability of an event, be it complex or simple. We shall try to comprehend the tools used for this purpose. In an attempt to make the subject easily comprehensible and clearly understandable a large number of illustrations and problems are selected and solved. Every aspect is fully supported by sufficient number of relevant examples. The students will note that for the understanding of the subject matter of this chapter, the knowledge of permutation and combination is essential.

We shall first throw some light on the meaning of probability and then proceed toward the various terms related to it.


Examples of Probability
Probably, it will rain today

It is possible that there will be drought in this area due to scanty rain

India is likely to win the world cup

In each of the above sentences there are different degrees of certainty of results.

This degree of certainty can be measured with the help of Probability.

Hence, Probability in general terms can be described how likely something will happen.