Pascal and the Probability Theory - Research Paper Example

? Pascal and the Probability Theory Biography of Pascal Blaise Pascal was a French monk, philosopher, physician, mathematician, writer and inventor. Born on June 19, 1623 in Auvergne, France, he died at a tender age of 39 in 1662. Despite living for a few years, Pascal was able to come up with many discoveries and theories in the fields of mathematics, philosophy, science and religion. His mother died when he was three; hence, he was raised by his father, who oversaw his education at home. His father introduced him to Mathematics when he was 15 years old. Despite this fact, he had started making discoveries in geometry as early as 12 years. Come 1939, his family moved to Rouen after his father was appointed a tax collector. His creativity was exhibited at this point, whereby; he was able to invent a calculator so as to make his father’s work easier during his day to day endeavors. Later in 1647, Pascal conducted and wrote about experiments concerning vacuums, but many scientists disregarded the existence of vacuums. For instance, the great French philosopher Rene Descartes visited Pascal for 2 days and disagreed with him concerning the vacuum. In a letter Descartes wrote to Huygens after this encounter, he asserts that “Pascal had too much vacuum in his head”1. This did not deter him from pursuing mathematics and science; in 1653, he published a treatise on the equilibrium of Liquids. Pascal later invented “Pascal’s Triangle”, a triangular array of numbers which solves mathematical problems. He would later become interested in probability after a gambling question was projected to him. In correspondence with another great scientist Fermat, they were able to solve the problem of points; how to divide stakes in a game of dice if the game is incomplete. Pascal almost lost his life in 1654, and this affected him psychologically to the point of being a staunch Christian. Even after setting the ground for the theory of probability, Pascal quit Mathematics to pursue a quieter Christain life in a monastery in Paris. He never married. He succumbed in 1662 to malignant stomach ulcers. Introduction Even from an early age, Pascal was curious about natural occurrences, and he formulated experiments to study these occurrences. Pascal studied fluids, pure and applied sciences, pressure and machines, among other scientific principles. While still a teenager in 1642, Pascal had begun ground-breaking work on calculating machines. At first, his attempts failed, but he finally managed to invent the mechanical calculator three years later2. This simply shows that Pascal was able to invent scientific principles from an earlier age. However, one of Pascal’s greatest invention is in the field of Mathematics is the “Theory of Probability”, a theory that is centered on estimating and calculating the chance of doing something. For example, if 6 men are playing dice, what is the chance that each of them can win the game? And if at all two are eliminated, what is the chance that each of the remaining four will win the game? Probability centers on such occurrences. The theory of probability was advanced so as to prove the fact that chance can be mathematically calculated accurately. Various facets in the modern world apply probability. This is not just in mathematics courses, but it is also applicable in the practical courses like genetics, quantum mechanics, kinetic theory of gases, industrial quality control and insurance.This paper dwells on the history of the probability theory, and how Pascal contributed to this theory, and how his findings were shaped by other great scientists who were also researching on the theory. Pascal’s involvement with probability dates back to 1654 when another scientist, Chevalier challenged him to solve a puzzle which at that time was known as “the problem of points”. This problem had been posed in the late 1400s and no one had managed to solve it for a period of 200 years. The problem was that; how the stakes of a chance game should be divided in case the game is not played to completion. This puzzle triggered Pascal to start thinking about probability immensely. He sought the correspondence of the then great Mathematician Pierre de Fermat. The work of these two scientists laid the foundation for the modern theory of probability. Pascal and Fermat realized that in solving the puzzle, all the possibilities needed to be listed, and then from this it would be possible to estimate the chance of each of the 6 players winning. In solving the puzzle, Pascal used the approach of tossing coins between two players. He used T to represent Tails and H to represent Heads. In ensuring that there is a winner, Pascal asserted that the coin will have to be tossed thrice. He then found out that in the three tosses, the combinations of tails and heads totaled 8. In case the first toss is Heads, Pascal listed the possibilities as follows; HHH, HHT, HTH and HTT. In this case, Player 1 wins for the first 3 instances, and player 2 wins on the last instance. Therefore, Pascal discovered that counting all possibilities first is the only way of ensuring the stakes of each player in case a game is interrupted abruptly. Pascal was able to invent more rules of probability. His findings formed the basis for the understanding of probability. History of probability; Discoveries of other mathematicians and how they relate to Pascal’s findings The doctrines of probability began to be used by people many years ago. There is no doubt that the elements of probability were useful during the census of population in ancient countries such as china, Egypt and India. The theory of probability came into life in the middle of XVII century in France. Probability theory is an element of mathematics that deals with influencing the long run chance that a specified incident will come about. This frequency is influenced by isolating the number of chosen proceedings by the total number of total actions possible. For instance, every six sides of a dice has one in six probabilities on a sole throw. Determined by the challenges experienced by seventeenth gamblers, this theory has grown into the most revered and most important faculties of mathematics with uses in many various activities. Conceivably, the basic thing that makes probability theory the nearly all priceless is that it can be applied to influence the anticipated results in any occurrence from the likelihood that a Cruise ship will capsize to the possibility that someone will win the sweepstake3. Probability concepts did not simply emerge out of nowhere; they have been in existence for thousands of years. In the 15th century, various works on probability started emerging. In 1494, an Italian by the name Fra Luca Paccioli advanced the first ever work on probability, Summa de arithmetica geometria proportioni e proportionalita. Later in 1550, another scientist, Geronimo Cardano continued Paccioli’s work by writing his own version called Liber de Ludo Aleau which translates to “A book on the Games of Chance”. In the 17th century, a question directed to Pascal culminated in the probability theory as we know it today. Chevalier de Mere was a nobleman who gambled a lot so as to increase his wealth. Initially, Chevalier bet always that the number 6 would appear in 4 rolls. He was tired with this approach as time went by, and he decided to change. However, he realized that his new approach of choosing 12 or a double 6 yielded less money than the old approach. He then asked Pascal why this happened. This kept Pascal thinking, and using probability, he discovered that the new approach had a 49.1% probability of winning when compared to the 51.8% of the old approach. Pascal did not solve this alone; Paris lawyer and mathematician, Pierre de Fermat assisted him in this. This marked the beginning of one of the greatest correspondences in mathematics. These two scientists continued to exchange their thoughts on mathematical problems and principles through letters. These two are credited with founding the theory of probability. Pascal asserted that probabilities can or cannot be reliant on each other. For instance, we might ask about the possibility of making a choice on a black card or queen from a deck of cards. These occasions are non reliant because even if you decide on a black card, you could still pick a queen. As a perfect illustration of conditional probability, think about a test in which one is permitted to select a ball out of a vase which holds five green balls and five black balls. On the first shot, a person would have an equivalent possibility of choosing either a green or a red ball. At this point, the likelihood of choosing one of the two colors is dissimilar on the second attempt, since only four balls of identical color can remain. Since time immemorial, gamblers have continued to rely on probability. Nevertheless, probability theory as of today has to a large extent a wider variety of uses. For instance, one of the great advancements that took physics by surprise, in the twenties, was the recognition that many occurrences in nature could not be explained with ideal conviction. Pascal favored another method the” method of expectations”. This method depends on doctrines of impartiality/fairness. This method effectively assisted Pascal to explain many principles that many scientists would not be able to explain at the time. But as long as the game is just, Pascal believed that both the gamblers have identical right to anticipate winning the next summit4. Assume, for instance, Patrick and Patterson have gambled equal money and being the first to concede four points, and they want to stop the competition when Patrick three points and Patterson lacks only two points. Patterson should receive more than Patrick because he is the one leading. If two more competitions were to be held, there would be four possibilities. Patterson wins the first and the second Patterson wins the first, and Peter wins the second Patrick wins the first and Patterson wins the second Patrick wins the fast and the second5 In the first three cases, Patterson wins the competition, and in the fourth scenario Patrick wins the game. Going by Cardano’s doctrine the gamble should be in equal quantity: three four Patterson and one for Patrick. If the gamble is equally divided, Patterson gets three-fourths6. Equally, if Patterson emerges as winner the whole gambling process, he wins the whole stakes. This grants him the privilege of having half of the stakes. If Patrick wins, the two are tied. Therefore, both are unconstrained to half of what has remained. Extending this unconventional thinking by mathematical stimulation, and applying the recursive elements of Pascal’s triangle, Pascal managed to answer the problem for every number of marks the players may be deficient in. He established, for instance, that if Patrick lacks four points and Patterson two points, then their go halves are established by summing the numbers in the foundation of the triangle. Patterson’s share is to Patrick’s as 1+5+10+10 to 5+17. Pascal organized much of his understanding, jointly with his universal elucidation in his book Traite du triangle arithmetique (Arithmetic Triangle) which was published later. Huygens after hearing about the ideas of Pascal was obliged to work out the particulars for himself. He only realized this by writing a treatise that essentially was informed on Pascal’s method of opportunities. However, Huygens ventured farther than Pascal. Bernoulli also seems to strengthen this plan with his acclaimed theorem that argues that it is honorably certain that the regularity of an occasion in a bigger number of tests will estimate its possibility. Bernoulli was determined to prove this theorem and gave barely an insufficiently bigger upper bounce on the number of tests required for ethical firmness that the regularity would be contained by a given aloofness of possibility8. On the other hand, Nicholas Bernoulli advanced the upper bounce, and De Moivre expertly approximated the number of tests required, using an assortment of development of the essential of what we at present call “the normal destiny”. De Moivre was of different opinion; he lacked our modern concept of a possibility circulation. In short, he was purely improving on Bernoulli in discovering the number of tests needed in helping to ascertain that the practical frequency would estimate the possibility9. When looking at Bernoulli’s theorem and his doctrines for assimilating possibilities, he failed to accomplish his ambition of making possibility an instrument for day to day life and official affairs. His doctrines for assimilating probabilities were argued in textbooks, and not used in observation. The mounting self-sufficiency of possibility from fairness in the early eighteenth century can be viewed as in complementary approaches of Nicholas Bernoulli and his relative Daniel Bernoulli. In the last fall the two mathematicians talked about the St. Petersburg problem10. A person, for instance, tosses a die frequently, and he triumphs in the competition when he primary gets a 6. The reward is doubled each time he does not succeed to have a 6. Equally, he is rewarded one crown if he acquires a 6 on the first toss, and vice versa11. Going by Huygens’s rules, he should at least forfeit an inestimable amount, and nevertheless; no one will be willing to part away with an extra crowns. Daniel Bernoulli expounded more on this by inventing the thought of “probable utility”. Nicholas Bernoulli was surely opposed to this explanation, because for him the hypothesis of possibility was primed on equity. Though possibility had established a hypothetical position in the mathematical theory in 1750, the usages of the hypothesis were equally just to the queries of equity. No particular person had studied how to apply probability in data examination. This was genuine still in the work on pension and life indemnity. De Moivre applied hypothetical transience curves. Equally, Simpson used the transience data; however nobody applied probabilistic techniques to replace the illustration in the way contemporary demographers do. Although convenient task of combining observations was not influenced by possibility hypothesis until subsequent to Legendre’s periodical of slight squares, efforts to prime the techniques for assimilating observations on possibility hypothesis began so early. One of the most significant achievements of probabilistic work on the discovery by Laplace was technique of inverse possibility; this discovery is nowadays known as the Bayesian technique of statistical inference. Laplace invented inverse possibility in the itinerary of his work on the hypothesis of errors in the mid 1770s. Laplace grasped that probabilities for errors, previously before the annotations are preset, interpret into possibilities for the mysterious quantities being probable. He named them subsequent probabilities, and he reasonably defended using them by accepting the doctrine that following surveillance, the possibilities of its probable causes are comparative to the possibilities specified by the observation of the sources. In the nineteenth century it was known as the technique of inverse probability. Inverse probability gained recognition by Laplace’s colleagues as very significant contribution to the hypothesis of probability. Laplace’s career began to take off during the 1770s and 1780s, at this time he was making great strides in advancing the arithmetical techniques for assessing posterior probabilities. But this task did not right away bring probability theory into touch with the realistic problem of assimilating observations. When applied to the miscalculation distributions projected by Laplace and his colleagues, inverse probability techniques for assimilating observations that were willful in association to the recognized techniques. Thereafter, this changed Legendre’s discovery of the slightest squares. In the fall of 1809, another mathematician by the name Carl Friedrich Gauss was of the belief that he was solely the architect of the technique, and published his own version of it12. Gauss offered a possibility validation for smallest amount of squares: these squares approximate of a measure are the worth with maximum posterior probability if the inaccuracies have what is commonly known as a normal distribution. A distribution with possibility compactness for a constant h. in essence, gave a rather weak explanation. It is the barely error circulation which makes the numerical mean main significance whenever we have comments of a solitary unknown measure. Consequently the agreement that prefers the numerical mean must be an agreement in rapport with the error of distribution13. Conclusion Even though Pascal is credited with forming the foundation for the theory of probability, he has been criticized because his analysis did not tackle realistic situations e.g. the weather. Pascal asserted that it is important to list all possible outcomes in a situation before determining the chance of each of these situations. However much this is true, it is not possible to list possible outcomes in real situations. Pascal’s work in probability was simply the beginning. His was just developing the general rules governing probability. He later abandoned Mathematic, and instead opted for a quiet life in a monastery in Paris. Other scientists, however, continued Pascal’s work, and through their research, they were able to seal any loopholes in Pascal’s theory. For example, Jacob Bernoulli was able to invent statistical sampling. This addressed the problem of having outcomes that were infinite, and would not be listed according to Pascal’s theory. Bernoulli was also able to prove the fact that; as possible observations increased, uncertainty decreased. Many more theories pertaining to uncertainty and risk have been advanced over the years; however, the earlier assertions of Pascal, De Moivre and Bernoulli form the basis for risk, uncertainty and probability. Bibliography Edwards, Anthony William Fairbank. Pascal's arithmetical triangle: the story of a mathematical idea. Maryland: JHU Press, 2008. Glenn Shafer, Vladimir Vovk. Probability and finance: it's only a game! New Jersey: John Wiley and Sons, 2009. Hald, Anders. A history of probability and statistics and their applications before 1750. New Jersey: Wiley-IEEE, 2009. Hoffman, Frederick. Mathematical aspects of artificial intelligence: American Mathematical Society short course, January 8-9, 1996, Orlando, Florida. New York: American Mathematical Soc., 2009. Shea, William R. Designing experiments & games of chance: the unconventional science of Blaise Pascal. New York: Science History Publications, 2009. Retrieved from http://www.glennshafer.com/assets/downloads/articles/article50.pdf. Retrieved on 3th April, 2012 Read More