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This paper was written as an assignment for Ian Walton's Math G - Mathfor liberal Arts Students - at Mission College. If you use material pleaseacknowledge it.
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BritneyMaracchini
October 22,2001
Math G
MW 5 – 7p.m.
Midterm/Probability
Probability is a ratio that tells us thelikelihood of a specific outcome over others. We have all seen probabilities,whether we have recognized them as such or not. An example of a probability isthe listing of odds of winning a contest, such as the Publisher’sClearinghouse Sweepstakes or the California Lotto. They list the number oftimes we can expect to win, followed by the number of times we can expect tolose. The numbers are usually very large, such as 1:1,000,000,000.
Probability theory developed in themid-1650’s. It is interesting to note that the theory of probabilitywould probably not have developed so early if it weren’t for thepopularity of gambling. A disagreement over what winnings a person should beable to claim in a particular game of dice should that person decide to quitrather than roll the dice again began a series of letters back and forthbetween Blaise Pascal and Pierre de Fermat.
Blaise Pascal was a Frenchman, born inClermont, France, on June 19, 1623. He had a fascinating childhood, being homeschooled by his father (or a tutor, depending upon the resource you read) andforbidden to study mathematics. In fact, some reports claim that his fatherthrew away all the mathematic books he owned to keep Blaise from beingoverworked. Others say just that his father did not want Blaise to studymathematics until he was older. Regardless, being a normal child, Blaise wasnaturally curious about mathematics since he was forbidden to study them.Again, depending on the resource accessed, Blaise was either able to sneak amathematics book from an acquaintance or began working on mathematics all byhimself. When his father discovered that Blaise was studying mathematics, hisfather saw his great aptitude for the subject and gave in, allowing Blaise topursue this discipline.
Blaise’s natural ability is welldocumented in his many accomplishments in youth. He mastered Euclid’sElements, a book his father gave him, bythe age of 12.[1]
“Atsixteen, Pascal wrote an essay on conic sections; and in 1641, at the age ofeighteen, he constructed the first arithmetical machine, an instrument which,eight years later, he further improved.”
He called his machine the Pascaline andinvented it in hopes that it would help his father, a tax collector, to moreeasily perform his job.
Mathematicswere not, however, Blaise’s only interest. At 14, he went to meetingswith his father that were heavily influenced by the religious order of theMinimis. Then, in 1946, his father had a severe injury and was forced to betaken care of in his home by two members of a religious order. Blaise spentmany hours in conversation with these “brothers” during hisfather’s recuperation. Combined with the religious teachings of theMinimis, these sessions so affected his life, that in 1650, he gave up hismathematical pursuits to immerse himself in religion full-time.
Only after his father’s death inSeptember of 1651 and the settling of the estate in 1653, did Blaise return towork on mathematics. During this spurt of curiosity, he “invented thearithmetical triangle, and together with Fermat created the calculus ofprobabilities.”
As with Blaise Pascal, information on Pierrede Fermat depends upon the resource accessed. He was born in France on August17, 1601, and was either home-schooled or educated in the local Franciscanmonastery. He did not seriously pursue mathematics until he attended theUniversity of Toulouse, where he “produced important work on maxima andminima”.[4] Heeventually moved to Bordeaux, where he completed his degree work in civil law.He then returned to Toulouse, where he devoted most of his spare time to hismathematical pursuits.
Although a bright and talentedmathematicians, Pierre was kind of quirky in that he did not publish any of hiswork during his lifetime, nor were his methods well documented. Much as anabsent-minded professor, most of his teachings were learned from the scribbledand crumpled notes found strewn among his possessions after his demise.
Pierre was particularly interested inproving geometrical theorems. He remained in frequent contact with otherscholars through written correspondence, and quickly gained a reputation forbeing one of the leading mathematicians in the world. This communication wasnot always friendly, however. Frequently he would tear apart another’stheories and make fast enemies.
There were similarities between the lives ofBlaise and Pierre. Like Blaise, Pierre also had life threatening experiences.He contracted the plague and “in 1653 his death was wrongly reported,then corrected:
Iinformed you earlier of the death of Fermat. He is alive, and we no longer fearfor his health, even though we had counted him among the dead a short timeago.”[5]
Similarly, Pierre’s mathematical workwas interrupted twice; once when his work became too burdensome to allow himleisure time and once during a civil war in France. But whatever theirlikenesses, perhaps it was just plain fate that prodded Pierre to write Blaisefor confirmation of his ideas on probability.
The French nobleman, Chevalier de Mere,provided the inspiration for probability theory. Obsessed with gambling, he hadbegun to question the common assumption that one should bet that a double sixwould be rolled in 24 rolls of a pair of dice. Further, he wondered how moneyat stake should be divided should equally skilled players stop playing a gamebefore the game is finished. Having a great appreciation for the mathematicalabilities of Blaise Pascal, he contacted Blaise for advice. Blaise, in turn,sought Pierre’s opinions. Thus began, in the summer of 1654, acorrespondence by letter between Blaise and Pierre. In this series of fiveletters, the general foundation for probability theory was established.
Although Blaise and Pierre agreed upon thesame answer to de Mere’s “equally skilled players problem”,they went about proving their answers different ways. Blaise gave the followingaccount of his proof:
“...when two players play a game of three points and each player has staked 32pistoles.
Supposethat the first player has gained two points, and the second player one point;they have now to play for a point on this condition, that, if the first playergain, he takes all the money which is at stake, namely, 64 pistoles; while, ifthe second player gain, each player has two points, so that there are on termsof equality, and, if they leave off playing, each ought to take 32pistoles....If therefore the players do not wish to play this game but toseparate without playing it, the first player would say to the second, “Iam certain of 32 pistoles even if I lose this game, and as for the other 32pistoles perhaps I will have them and perhaps you will have them; the chancesare equal. Let us then divide these 32 pistoles equally, and five me also the32 pistoles of which I am certain.” Thus the first player will have 48pistoles and the second 16 pistoles.
Next,suppose that the first player has gained two points and the second player none,and that they are about to play for a point; the condition then is that, if thefirst player gain this point, he secures the game and takes the 64 pistoles,and if the second player gain this point, then the players will be in thesituation already examined, in which the first player is entitled to 48pistoles and the second to 16 pistoles. Thus if they do not wish to play, thefirst player would say to the second, ”If I gain the point I gain 64pistoles; if I lose it, I am entitled to 48 pistoles. Give me then the 48pistoles of which I am certain, and divide the other 16 equally, since ourchanges of gaining the point are equal.” Thus the first player will have56 pistoles and the second player 8 pistoles.
Finally,suppose that the first player has gained one point and the second player none.If they proceed to play for a point, the condition is that, if the first playergain it, the players will be in the situation first examined, in which thefirst player is entitled to 56 pistoles; if the first player lose the point,each player has the a point, and each is entitled to 32 pistoles. Thus, if theydo not wish to play, the first player would say to the second, “Give methe 32 pistoles of which I am certain, and divide the remainder of the 56pistoles equally, that is divide 24 pistoles equally.” Thus the firstplayer will have the sum of 32 and 12 pistoles, that is, 44 pistoles, andconsequently the second will have 20 pistoles.””
While lengthy, his answer is easy tounderstand and follow. Simply put, one is always entitled to what one hasalready, plus ½ of what remains because there is an equal chance ofwinning and losing at each turn.
PierreFermat attacked the problem in a different manner. He worked with combinations.This methodology is illustrated in the following excerpt from a letter datedAugust 24, 1654. Person A and person B are playing a game. Person A needs justtwo more points to win; B needs three points.
“Thenthe game will be certainly decided in the course of four trials. Take theletters a and b, and write down all the combinations that can be formed of fourletters. These combinations are 16 in number, namely, aaaa, aaab, aaba,aabb, abaa, abab, abba, abbb; baaa, baab, baba, babb; bbaa, bbab, bbba, bbbb
Pierre’s explanation is not so lengthy,but just as easy to follow. In both answers, the fundamental probability theoryemerges. The probability is equal for all outcomes, but the trick is todetermine what all the probable outcomes are, then calculate how many arefavorable to the one desired.
It is from these very simplistic beginningsthat today’s probability applications have grown. Jacob Bernoulli,(1654-1705), built the basis of mathematical statistics (applied probability)in his trials on repeated experiments. Since certain outcomes were designatedsuccesses and others failures, Bernoulli showed that the probability of successmust remain the same from experiment to experiment. This can be extended tobinomial probabilities if a random variable is first defined and then aprobability function of a random variable.
Probabilities are also very useful in thefield of genetics for humans and other life sciences. When one knows the makeup of two items, be they human or nature, and they can look at all possiblecombinations of the two to predict the probability of offspringcharacteristics. Genetic counseling uses this type of probability combined withother risk factors, such as age of the perspective parents.
Probability also has applications inpsychology. This is especially true in soothing today’s society that isstill reeling from the events of September 11, 2001. Anthrax has scooped theheadlines and false alarms are being called in to the police departments at analarming rate. As of this writing, though, there are only approximately 12confirmed cases of anthrax in the United States, which has a population of some300 million people (12 in 300,000,000). The probability of one of us beingexposed and contracting it, then, is much less than the probability of ourbeing struck by lightning (1 in 709,260). Such information should help to sooththe psychological effects of the recent terrorism.
Probability has a place in economics aswell. This is called the mathematical expectation. Introduced by Huygensin1657, it shows that “if p is theprobability of a person winning a sum a, and q that of winning asum b, then he may expect to winthe sum ap + bq
Thus has been the effect of the simple calculationsof a 17th century gambler and his quest to determine whether thecommonly accepted rule that “betting on a double six in 24 throws wouldbe profitable “
This paper was written as anassignment for Ian Walton's Math G - Math for liberal Arts Students - atMission College. If you use material please acknowledge it.
Bibliography
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Ball, W. W.Rouse. Blaise Pascal (1623-1662).
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Eves, Howard. AnIntroduction to the History of Mathematics. Saunders College Publishing, Chicago. 1990. pp. 328, 331,357-358, 362, 428-429, 446, 518, 572
O’Connor,J. J. and Robertson, E.F. Blaise Pascal.
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Footnotes