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This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material please acknowledge it.
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Britney Maracchini
October 22, 2001
Math G
MW 5 – 7 p.m.
Midterm/Probability
Probability is a ratio that tells us the likelihood 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 is the listing of odds of winning a contest, such as the Publisher’s Clearinghouse Sweepstakes or the California Lotto. They list the number of times we can expect to win, followed by the number of times we can expect to lose. The numbers are usually very large, such as 1:1,000,000,000.
Probability theory developed in the mid-1650’s. It is interesting to note that the theory of probability would probably not have developed so early if it weren’t for the popularity of gambling. A disagreement over what winnings a person should be able to claim in a particular game of dice should that person decide to quit rather than roll the dice again began a series of letters back and forth between Blaise Pascal and Pierre de Fermat.
Blaise Pascal was a Frenchman, born in Clermont, France, on June 19, 1623. He had a fascinating childhood, being home schooled by his father (or a tutor, depending upon the resource you read) and forbidden to study mathematics. In fact, some reports claim that his father threw away all the mathematic books he owned to keep Blaise from being overworked. Others say just that his father did not want Blaise to study mathematics until he was older. Regardless, being a normal child, Blaise was naturally curious about mathematics since he was forbidden to study them. Again, depending on the resource accessed, Blaise was either able to sneak a mathematics book from an acquaintance or began working on mathematics all by himself. When his father discovered that Blaise was studying mathematics, his father saw his great aptitude for the subject and gave in, allowing Blaise to pursue this discipline.
Blaise’s natural ability is well documented in his many accomplishments in youth. He mastered Euclid’s Elements, a book his father gave him, by the age of 12.[1]
“At sixteen, Pascal wrote an essay on conic sections; and in 1641, at the age of eighteen, he constructed the first arithmetical machine, an instrument which, eight years later, he further improved.” [2]
He called his machine the Pascaline and invented it in hopes that it would help his father, a tax collector, to more easily perform his job.
Mathematics were not, however, Blaise’s only interest. At 14, he went to meetings with his father that were heavily influenced by the religious order of the Minimis. Then, in 1946, his father had a severe injury and was forced to be taken care of in his home by two members of a religious order. Blaise spent many hours in conversation with these “brothers” during his father’s recuperation. Combined with the religious teachings of the Minimis, these sessions so affected his life, that in 1650, he gave up his mathematical pursuits to immerse himself in religion full-time.
Only after his father’s death in September of 1651 and the settling of the estate in 1653, did Blaise return to work on mathematics. During this spurt of curiosity, he “invented the arithmetical triangle, and together with Fermat created the calculus of probabilities.”[3] Fate, however, intervened in his life once more. As he was driving a horse-drawn carriage, his horses bolted over a bridge, leaving the carriage precariously dangling above a river. This potentially life threatening experience renewed his religious fervor, and he dedicated the rest of his life to Christianity. Frequently visiting the Jansenist monastery, Blaise wrote many theological works. His final years were dedicated to caring for the poor and the church. He died physically worn out from insomnia and acute dyspepsia, and suffering from a malignancy.
As with Blaise Pascal, information on Pierre de Fermat depends upon the resource accessed. He was born in France on August 17, 1601, and was either home-schooled or educated in the local Franciscan monastery. He did not seriously pursue mathematics until he attended the University of Toulouse, where he “produced important work on maxima and minima”.[4] He eventually 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 his mathematical pursuits.
Although a bright and talented mathematicians, Pierre was kind of quirky in that he did not publish any of his work during his lifetime, nor were his methods well documented. Much as an absent-minded professor, most of his teachings were learned from the scribbled and crumpled notes found strewn among his possessions after his demise.
Pierre was particularly interested in proving geometrical theorems. He remained in frequent contact with other scholars through written correspondence, and quickly gained a reputation for being one of the leading mathematicians in the world. This communication was not always friendly, however. Frequently he would tear apart another’s theories and make fast enemies.
There were similarities between the lives of Blaise and Pierre. Like Blaise, Pierre also had life threatening experiences. He contracted the plague and “in 1653 his death was wrongly reported, then corrected:
I
informed you earlier of the death of Fermat. He is alive, and we no longer fear
for his health, even though we had counted him among the dead a short time
ago.”[5]
Similarly, Pierre’s mathematical work was interrupted twice; once when his work became too burdensome to allow him leisure time and once during a civil war in France. But whatever their likenesses, perhaps it was just plain fate that prodded Pierre to write Blaise for confirmation of his ideas on probability.
The French nobleman, Chevalier de Mere, provided the inspiration for probability theory. Obsessed with gambling, he had begun to question the common assumption that one should bet that a double six would be rolled in 24 rolls of a pair of dice. Further, he wondered how money at stake should be divided should equally skilled players stop playing a game before the game is finished. Having a great appreciation for the mathematical abilities of Blaise Pascal, he contacted Blaise for advice. Blaise, in turn, sought Pierre’s opinions. Thus began, in the summer of 1654, a correspondence by letter between Blaise and Pierre. In this series of five letters, the general foundation for probability theory was established.
Although Blaise and Pierre agreed upon the same answer to de Mere’s “equally skilled players problem”, they went about proving their answers different ways. Blaise gave the following account of his proof:
“... when two players play a game of three points and each player has staked 32 pistoles.
Suppose that 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 player gain, he takes all the money which is at stake, namely, 64 pistoles; while, if the second player gain, each player has two points, so that there are on terms of equality, and, if they leave off playing, each ought to take 32 pistoles....If therefore the players do not wish to play this game but to separate without playing it, the first player would say to the second, “I am certain of 32 pistoles even if I lose this game, and as for the other 32 pistoles perhaps I will have them and perhaps you will have them; the chances are equal. Let us then divide these 32 pistoles equally, and five me also the 32 pistoles of which I am certain.” Thus the first player will have 48 pistoles 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 the first 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 the situation already examined, in which the first player is entitled to 48 pistoles and the second to 16 pistoles. Thus if they do not wish to play, the first player would say to the second, ”If I gain the point I gain 64 pistoles; if I lose it, I am entitled to 48 pistoles. Give me then the 48 pistoles of which I am certain, and divide the other 16 equally, since our changes of gaining the point are equal.” Thus the first player will have 56 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 player gain it, the players will be in the situation first examined, in which the first 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 they do not wish to play, the first player would say to the second, “Give me the 32 pistoles of which I am certain, and divide the remainder of the 56 pistoles equally, that is divide 24 pistoles equally.” Thus the first player will have the sum of 32 and 12 pistoles, that is, 44 pistoles, and consequently the second will have 20 pistoles.””[6]
While lengthy, his answer is easy to understand and follow. Simply put, one is always entitled to what one has already, plus ½ of what remains because there is an equal chance of winning and losing at each turn.
Pierre Fermat attacked the problem in a different manner. He worked with combinations. This methodology is illustrated in the following excerpt from a letter dated August 24, 1654. Person A and person B are playing a game. Person A needs just two more points to win; B needs three points.
“Then the game will be certainly decided in the course of four trials. Take the letters a and b, and write down all the combinations that can be formed of four letters. These combinations are 16 in number, namely, aaaa, aaab, aaba, aabb, abaa, abab, abba, abbb; baaa, baab, baba, babb; bbaa, bbab, bbba, bbbb. Now every combination in which a occurs twice or oftener represents a case favourable to A, and every combination in which b occurs three times or oftener represents a case favourable to B. Thus, on counting them, it will be found that there are 11 cases favourable to A, and 5 cases favourable to B; and since these cases are all equally likely, A’s chance of winning the game is to B’s chance as 11 is to 5.”[7]
Pierre’s explanation is not so lengthy, but just as easy to follow. In both answers, the fundamental probability theory emerges. The probability is equal for all outcomes, but the trick is to determine what all the probable outcomes are, then calculate how many are favorable to the one desired.
It is from these very simplistic beginnings that 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 designated successes and others failures, Bernoulli showed that the probability of success must remain the same from experiment to experiment. This can be extended to binomial probabilities if a random variable is first defined and then a probability function of a random variable.[8] This can further relate to market analyses, using approximate empirical probabilities to predict target markets and saturation levels.
Probabilities are also very useful in the field of genetics for humans and other life sciences. When one knows the make up of two items, be they human or nature, and they can look at all possible combinations of the two to predict the probability of offspring characteristics. Genetic counseling uses this type of probability combined with other risk factors, such as age of the perspective parents.
Probability also has applications in psychology. This is especially true in soothing today’s society that is still reeling from the events of September 11, 2001. Anthrax has scooped the headlines and false alarms are being called in to the police departments at an alarming rate. As of this writing, though, there are only approximately 12 confirmed cases of anthrax in the United States, which has a population of some 300 million people (12 in 300,000,000). The probability of one of us being exposed and contracting it, then, is much less than the probability of our being struck by lightning (1 in 709,260). Such information should help to sooth the psychological effects of the recent terrorism.
Probability has a place in economics as well. This is called the mathematical expectation. Introduced by Huygens in1657, it shows that “if p is the probability of a person winning a sum a, and q that of winning a sum b, then he may expect to win the sum ap + bq.”[9]
Thus has been the effect of the simple calculations of a 17th century gambler and his quest to determine whether the commonly accepted rule that “betting on a double six in 24 throws would be profitable “[10] was indeed a mathematically sound one.
This paper was written as an assignment for Ian Walton's Math G - Math for liberal Arts Students - at Mission College. If you use material please acknowledge it.
Bibliography
Apostol, Tom M. A Short History of Probability. Calculus, Volume II http://www.cc.gatech.edu/classes/c6751_967_winter/Topics/stat-meas/probHist.html. 10/08/01
Ball, W. W. Rouse. Blaise Pascal (1623-1662). http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Pascal.html
10/07/01
Ball, W. W. Rouse. Pierre de Fermat (1601-1665). http://www.maths.tcd.ie/pub/HistMath/People/Pascal/RouseBall/RB_Fermat.html
10/07/01
Barnett, Raymond A., Burke, Charles J., and Ziegler, Michael R. Applied Mathematics for Business and Economics, Life Sciences, and Social Sciences. Dellen Publishing Company, Santa Clara. 1983. pp. 427-428, 438-439.
Blaise Pascal. http://obiwan.stmarytx.edu/cspeople/pasca/pascal.htm 10/05/01
Eves, Howard. An Introduction 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.
http://www-groups.dcs.st-and.ac.uk:80/~history/Mathematicians/Pascal.html. 10/09/01
O’Connor, J. J. and Robertson, E.F. Pierre de Fermat.
http://www-groups.dcs.st-and.ac.uk:80/~history/Mathematicians/Fermat.html. 10/09/01
Percent and Probability. Math League Multimedia. http://www.mathleague.com/help/percent/percent.htm 10/09/01
Wheeler, Ruric. Finite Mathematics for Business and the Social and Life Sciences; A Problem-Solving Approach. Saunders College Publishing, Chicago. 1991. pp. 256, 285, 293, 301.
Footnotes