In the physical world, events occur according to natural laws within a realm of seeming randomness. For instance, a tree grows its leaves in a strict sequence of predictable biochemical processes which can be studied and described in minute detail. However, when the season has passed and the leaves die, there is no way to predict with perfect certainty when or where any individual leaf will fall and strike the ground. Nor can it be predicted with certainty whether a coin flipped into the air will land heads up or tails up. The laws of physics can precisely describe the dynamic forces that are acting on a spinning coin as it is flipped into the air. Indeed, it may well be that if one could account for every minute force, energy vector, and motion that affects a tumbling coin one could predict exactly, according to the principle of determinism, which side of the spinning coin will land facing up. The difficulty with the deterministic approach is that an overwhelming quantity of information is required in order to make such a prediction. The amount of data required to analyze the moment-by-moment fluctuations in the dynamics of a spinning coin are so great that the solution to finding the position of the coin's final resting place may be impossible to determine. Compound the problem by attempting to predict the final resting position of a thousand coins tossed into the air at once.
Rather than struggle to precisely analyze the exact behavior of each individual participant in a complex situation, such as a multiple coin toss, there exists a more practical method. Probability theory is a mathematical tool that enables scientists to study the behavior and consequences of complex situations--for example, to determine how often or how many times an event is likely to occur over a period of time. Probability theory makes it possible to predict large-scale events for which full knowledge would be impossible to obtain.
A vague, intuitive concept of chance, or "fate," has existed since ancient times, in which phenomena such as the path a rolling boulder follows as it tumbles down a hillside or the exact moment a leaf falls are considered to be undeterminable quantities and represent events that are controlled by cosmic forces which cannot be understood. This view began to change with the coming of the Renaissance in Europe which saw rapid progress in the development of algebra, analytic geometry and higher mathematics in general. In the sixteenth century, an eclectic Italian philosopher, physician, and mathematician named Girolamo Cardano presented the idea that, beyond luck, rules of chance governed the outcome of undecidable events. He developed mathematical expressions of this idea but, unfortunately, never published his results. The full development of probability theory had to wait for many more decades.
Around 1654, the French mathematician Blaise Pascal, in a celebrated correspondence with the amateur mathematician Pierre de Fermat, laid down the foundations of the calculus of probability by creating mathematical techniques for determining the likelihood of favorable outcomes in games of chance such as dice. The detailed analysis of games, which presented clearly defined and abstract situations, became an important method for probing the properties of randomness. In fact, owners of gambling houses and casinos hire mathematicians to use the latest techniques of probability theory to calculate the risk of loss to the house in various popular games. Insurance companies also needed realistic and accurate statistics to devise saleable insurance policies. Actuarial tables, which compared the frequency of the deaths of individuals to their ages, were a hot commodity in the seventeenth century and were provided to insurance companies by mathematicians such as Edmond Halley and Abraham de Moivre, who developed a binomial probability distribution which related "chances" to probability.
Before 1711 only Christiaan Huygens and de Montmort had published texts on probability, but with the development of the calculus in Britain and Europe during the seventeenth century, scientists openly began to tackle complex problems in probability. Notable among these scientists were the Swiss mathematicians Jean Bernoulli and Jacques Bernoulli. The Bernoulli brothers furthered the acceptance of Leibnizian calculus in Europe and used it to develop an elaborate theory of probability. Jacques was the first mathematician to rigorously link the chance of an event occurring to its observable frequency.
More than just calculating the odds of winning at poker or the risk of dying after age 40, the techniques of probability theory were needed to help study the complex phenomena of nature. The increasing study of seemingly random events involving magnetism and electric charges benefitted from the application of an understanding of the laws of chance. A full flowering of the principles of probability took place in the nineteenth century with the work of Siméon-Denis Poisson and Jules Poincaré among others.
In the early twentieth century, quantum theory evolved as the most satisfactory model for explaining the behavior of the atom and today represents one of the most far reaching and important applications of probability theory. Further work on modern probability theory by mathematicians such as the Russian Andrei Markov and Austrian-American Richard von Mises has made it possible to study atomic particle interactions, crystal structure and exceedingly complex statistical problems such as the behavior of commercial telephone traffic, the landing of aircraft and the forecasting of economic trends.
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