# Probability

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**Probability**, also known as ‘the theory of probability’, is the branch of mathematics that deals with the study of random phenomena. The theory of probability allows us to quantitatively measure the degree of uncertainty involved in such cases and forms the backbone of modern statistics.

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## History

One of the first attempts at a quantitative measure of probability was taken in the 16th century by the Italian scientist Galilee Galileo while studying a problem relating to gambling. The foundation of the modern theory of probability occurred in the seventeenth century when two French mathematicians Blaise Pascal and Pierre de Fermat began to analyze dice problems posed by the French gambler Chevalier-De-Mere.

Jakob Bernoulli’s posthumous publication, ‘Treatise on probability’, in the early 18th century was one of the most significant early contributions to this field. Another major contribution was made by the French mathematician Abraham de-Moivre, who wrote the book ‘Doctrine of Chances’ (1718). Other early contributors to this field include T. Bayes (inverse probability) and P. S. Laplace.

Many 19th century contributions were from Russia, including P. L. Chebyshev, who established the Russian school of statistics, A. Markoff (d.1922), A. Khintchine (Law of Large Numbers), Liapounoff (Central Limit Theorem) and A. Kolmogorov.

## Explanation

The probability of an event (or 'happening') is mathematically defined as the ratio of favourable outcomes (to the happening of the event) to the total number of exhaustive outcomes, provided all the outcomes are equally likely and mutually exclusive. For example, in the rolling of an unbiased dice, the probability that either a four or a six turns up is given by

The probability of an event is also defined empirically as the limiting value of the ratio of the number of times the event occurs to the total number of trials, provided that the trials are repeated in identical and consistent conditions. For example, consider a random experiment of a biased dice which is rolled a large number of times, say 100, and the number of fours and sixes observed are 50, then by the empirical definition, we have

## Properties of probability

The following are some of the fundamental properties of probability:

- The total probability for a given sample space equals one. The term ‘sample space’ refers to the set of all possible outcomes.
- The probability for an event always lies between zero and one. An impossible event has a probability of zero and a certain event has a probability of one.
- If A and B are two disjoint events, i.e. they do not have any common element, then the probability that both A and B will occur is equal to the probability of event A added to the probability of event B.
- The probability that an event will not occur is equal to one minus the probability that it will occur.
- If the event A is a subset of B, then the probability of A will always be either less than or equal to the probability of B.

## Applications of probability

Probability forms the basis of almost all modern developments in statistics, including distribution theory, testing of hypotheses, analysis of variance and design of experiments. All these sub-fields of statistics are used extensively in every sphere of human knowledge. Probability is required due to the need for a scientific approach in order to determine the degree of uncertainty involved while taking decisions.