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ADDITIONAL MATHEMATICS PROJECT WORK 2 2010 PART 1 a) History of probability The scientific study of probability is a modern development. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, but exact mathematical descriptions of use in those problems only arose much later. According to Richard Jeffrey, Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, univocally
  ADDITIONAL MATHEMATICS PROJECT WORK 22010  PART 1 a)   History of prob a bility The scientific study of probability is a modern development.Gamblingshows that there has beenan interest in quantifying the ideas of probability for millennia, but exact mathematicaldescriptions of use in those problems only arose much later.According to Richard Jeffrey, Before the middle of the seventeenth century, the term 'probable'(Latin  probabilis ) meant approvable , and was applied in that sense, univocally, to opinion and toaction. A probable action or opinion was one such as sensible people would undertake or hold, inthe circumstances. However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.Aside from some elementary considerations made byGirolamo Cardanoin the 16th century, thedoctrine of probabilities dates to the correspondence of Pierre de FermatandBlaise Pascal  (1654).Christiaan Huygens(1657) gave the earliest known scientific treatment of the subject.Jakob Bernoulli's   Ars Conjectandi (posthumous, 1713) andAbraham de Moivre's   Doctrine of Chances (1718) treated the subject as a branch of mathematics. SeeIan Hacking's  The Emergence of Probability andJames Franklin's  The Science of Conjecture for histories of theearly development of the very concept of mathematical probability.The theory of errors may be traced back toRoger Cotes's  Opera Miscellanea (posthumous,1722), but a memoir prepared byThomas Simpsonin 1755 (printed 1756) first applied the theoryto the discussion of errors of observation. The reprint (1757) of this memoir lays down theaxioms that positive and negative errors are equally probable, and that there are certainassignable limits within which all errors may be supposed to fall; continuous errors are discussedand a probability curve is given.Pierre-Simon Laplace(1774) made the first attempt to deduce a rule for the combination of observations from the principles of the theory of probabilities. He represented the law of  probability of errors by a curve  y = (  x ),  x being any error and  y its probability, and laid downthree properties of this curve:1.   it is symmetric as to the  y -axis;2.   the  x -axis is anasymptote, the probability of the error being 0;3.   the area enclosed is 1, it being certain that an error exists.He also gave (1781) a formula for the law of facility of error (a term due to Lagrange, 1774), butone which led to unmanageable equations.Daniel Bernoulli(1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.  ADDITIONAL MATHEMATICS PROJECT WORK 22010  PART 1 Themethod of least squaresis due toAdrien-Marie Legendre(1805), who introduced it in his  N  ouvelles méthodes pour la détermination des orbites des comètes (  N  ew Methods for  Determining the Orbits of Comets ). In ignorance of Legendre's contribution, an Irish-Americanwriter,Robert Adrain, editor of The Analyst (1808), first deduced the law of facility of error, h being a constant depending on precision of observation, and c a scale factor ensuring that thearea under the curve equals 1. He gave two proofs, the second being essentially the same asJohnHerschel's(1850).Gaussgave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823),James Ivory(1825, 1826), Hagen (1837),Friedrich Bessel(1838),W. F. Donkin(1844, 1856), andMorgan Crofton(1870). Other contributors were Ellis (1844),De Morgan(1864),Glaisher   (1872), andGiovanni Schiaparelli(1875). Peters's (1856) formula for  r  , the probable error of asingle observation, is well known.In the nineteenth century authors on the general theory includedLaplace,Sylvestre Lacroix  (1816), Littrow (1833),Adolphe Quetelet(1853),Richard Dedekind(1860), Helmert (1872), Hermann Laurent(1873), Liagre, Didion, andKarl Pearson.Augustus De MorganandGeorge Booleimproved the exposition of the theory.Andrey Markovintroduced the notion of Markov chains(1906) playing an important role in theory of stochastic processesand its applications.The modern theory of probability based on themeausure theorywas developed byAndrey Kolmogorov(1931).On the geometric side (seeintegral geometry) contributors to The Educational Times wereinfluential (Miller, Crofton, McColl, Wolstenholme, Watson, andArtemas Martin).  ADDITIONAL MATHEMATICS PROJECT WORK 22010  PART 1   a)   Prob a bility in our livesi )   We a ther forec a sting Suppose you want to go on a picnic this afternoon, and the weather report says that the chance of rain is 70%? Do you ever wonder where that 70% came from?Forecasts like these can be calculated by the people who work for the National Weather Servicewhen they look at all other days in their historical database that have the same weather characteristics (temperature, pressure, humidity, etc.) and determine that on 70% of similar daysin the past, it rained.As we've seen, to find basic probabilitywe divide the number of favorable outcomes by the totalnumber of possible outcomes in our sample space.If we're looking for the chance it will rain,this will be the number of days in our database that it rained divided by the total number of similar days in our database. If our meteorologist has data for 100 days with similar weather conditions (the sample space and therefore the denominator of our fraction), and on 70 of thesedays it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%.Since a 50% probability means that an event is as likely to occur as not, 70%, which is greater than 50%, means that it is more likely to rain than not. But what is the probability that it won't   rain? Remember that because the favorable outcomes represent all the possible ways that anevent can occur, the sum of the various probabilities must equal 1 or 100%, so 100% - 70% =30%, and the probability that it won't rain is 30%. ii )   Ba tting a ver a ges Let's say your favorite baseball player is batting 300. What does this mean?A batting average involves calculating the probability of a player's getting a hit. The samplespace is the total number of at-bats a player has had, not including walks. A hit is a favorableoutcome. Thus if in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For  baseball stats we multiply all the percentages by 10, so a 30% probability translates to a 300 batting average.This means that when a Major Leaguer with a batting average of 300 steps up to the plate, he hasonly a 30% chance of getting a hit - and since most batters hit below 300, you can see how hardit is to get a hit in the Major Leagues!  ADDITIONAL MATHEMATICS PROJECT WORK 22010  PART 1   a)   Introduction
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