a brief look at the history of probability and statistics

A Brief Look at the History of Probability and StatisticsAuthor(s): JAMES E. LIGHTNERSource: The Mathematics Teacher, Vol. 84, No. 8 (NOVEMBER 1991), pp. 623-630Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27967334 .

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A Brief Look at the History of

Probability and Statistics By JAMES E. LIGHTNER

The

NCTM's Curriculum and Evaluation Standards for School Mathematics

(1989) includes concepts from probability and statistics throughout the school pro

gram, K-12. Indeed, these areas of mathe

matics are listed as topics for increased at

tention in the curriculum. Because of this

emphasis, all teachers should become more

aware of the historical development of these

areas of mathematics so we can incorporate historical topics for motivation and exten

sion of the content and so we can show the

very human factor in the creation of math

ematics. Let us, therefore, trace the develop ment of the areas of probability and statis

tics.

Probability

Probability has been defined as the study of the frequency of appearance of a phenome non in relation to all possible alternatives.

According to some historians of mathemat

ics, probability theory began as a branch of

mathematics with the famous correspon dence between Blaise Pascal and Pierre de

Ferm?t in 1654. From another point of view, this explanation is quite wrong. Centuries

before Pascal and Ferm?t ever thought about defining probability, or the "problem of points," various probabilistic problems had been addressed by both lay people and

mathematicians. The difficulty in trying to

trace the origins of probability is that it

began essentially as an empirical science

and developed only much later as a mathe

James Lightner teaches mathematics at Western Mary land College, Westminster, MD 21157. His professional interests are in geometry and the history of mathematics,

as well as mathematics education. He also serves as

executive secretary of the Maryland Council of Teachers

of Mathematics.

matical science. Indeed, probability has twin

roots in two fairly different lines of thought: the solution of gambling, or betting, prob lems and the handling of statistical data

related to such quantitative instruments as

mortality tables and insurance rates. Let us

first examine the wagering problems of gam

bling.

Randomness was not associated with gaming in early times.

Did gambling develop from game play

ing, did it arise from religious activity, or

did it arise from wagering? No one knows.

We do know that by about 1200 b.c., cubical

marked dice had evolved from much cruder

bones (such as the astragalus in the foot

because of its shape) as a useful device for

randomization in games. The faces of the die

had been formed by grinding down the bone

into a rough cube; the faces were then

marked by drilling into them various num

bers of shallow depressions, probably be cause no standard or simple symbol for num

bers had yet been devised. The present arrangement of pips, with various combina

tions of seven on opposite faces, was intro

duced about 1400 b.c.

Games of chance are probably as old as

the human desire to get something for noth

ing. A board game called "hounds and jack als" was played in Egypt as early as 3500

b.c.; counters were moved according to cer

tain rules by throwing the various astragali. Herodotus wrote in the fifth century b.c. in

his History (1952, 2) that to adapt to the

problems of a famine in 1500 b.c., a group of

people gathered together dice and astragali

November 1991 623



and engaged "in games one day so entirely as not to feel any craving for food, and the next day to eat and abstain from games. In this way they passed eighteen years." We know also that the Romans had a passion for

gambling with dice. It is reported that the

emperor Claudius (10 b.c.-a.d. 54) was so devoted to dice that he published a book titled How to Win at Dice. Claudius even

played while being driven about, throwing dice on a board that had been fitted to his carriage.

Gaming seems to have reached such a level of popularity with the Greeks and Ro

mans that it eventually became necessary to forbid it legally except in certain seasons.

Later, during the Middle Ages, the Chris tian Church launched a campaign against playing with dice and cards, not so much because of the gambling, however, but rather because of the vices of drinking and

swearing that seemed to accompany the

gambling. During the Third Crusade (1190), no person below the rank of knight was al lowed to gamble for money, whereas knights and clergy, interestingly enough, could play but were not allowed to lose more than

twenty shillings in a twenty-four-hour pe riod. Medieval history is full of such at

tempts to prohibit or limit gambling. One might assume that during these sev

eral thousand years of dice-playing, some elements of a probability theory would have

begun to appear. Yet no direct link between

gambling and mathematics seems to have been observed. Apparently no one consid ered that calculating the frequency of falls of dice was possible or fruitful or even that each face would turn up with equal fre

quency. This inattention may have been

due, of course, to the lack of a perfectly balanced or "honest" die, which precluded any noticeable regularity. Or perhaps the absence of appropriate mathematical nota tion and symbols hampered investigations. A more powerful reason, however, might be that the concept of "randomness" itself was

contrary to the thinking of the time. It was believed that God, or many gods, directed

earthly events in some predetermined plan, and randomness was just not possible or

even considered. Hence, any substantial ap proach to the examination and calculation of random events did not take place until the

Renaissance, when the ability to write and calculate with Hindu-Arabic numerals had become widespread, simple algebra had been developed, and philosophical thinking had begun to change and broaden.

Life's important questions deal with probability.

The first true mathematical treatment of

probability began in the latter part of the fifteenth century and the early part of the sixteenth century, when some Italian math ematicians began to consider the mathemat ical chances in certain gambling games, in

cluding dice. Girolamo Cardano (Jerome Cardan) (1501-1576) who was rich in genius and often devoid of principle, was an Italian

professor of mathematics and medicine with a most interesting and varied career. His most famous work is the Ars Magna (Car dano 1968), originally published in 1545, in which he presented all the rules of algebra as far as they had been developed, including various methods of solutions to cubic and

quartic equations. Also, over a period of

forty years, Cardano gambled daily. Early in his life he determined that if one did not

play for monetary stakes, no compensation would be gained for the time lost in gam bling, which could otherwise be spent in such more worthwhile pursuits as learning. Since he did not wish to waste his time in

unprofitable activities, he seriously ana

lyzed the probabilities of drawing aces out of a deck of cards and of throwing sevens with two dice. Then he reported the results of these investigations, as well as his practical experiences, in a gambler's manual called Liber de Ludo Aleae (The gambling scholar) (1953), first published in 1539. To aid his fellow gamblers, he noted, for example, that when cutting a deck the chance of obtaining a certain card is considerably increased by first rubbing the card with soap! In a chapter titled "On the Cast of One Die," he reported,

624 Mathematics Teacher



"I am as able to throw 1, 3 or 5 as 2, 4 or 6.

The wagers are therefore laid in accordance with this equality if the die is honest and, if not, they are made so much larger or smaller

in proportion to the departure from true

equality" (Burton 1985, 418). Here for the

first time, we note the transition from em

piricism to the theoretical probabilistic con

cept of an honest die. In making this state

ment, Cardano could logically be considered

the father of probability theory. Thus was

founded the branch of mathematics that we

call probability. But at the time it may have

been too "gamey" for the mathematicians

and too mathematical for the gamblers! Historians generally agree that the

search for the answer to one very special

problem can be credited with giving rise to

the science of mathematical probability. This problem is the so-called "problem of

points," which deals with the division of the

stakes of a game of chance between two

equally skilled players when the game is

interrupted prior to its completion; the re

sult is based on the scores or "points" of the

players at the time of the interruption. Luca

Pacioli (1445-1509), in his Summa de Arit

metica of 1494, was one of the first writers to

present the problem of points in a mathe

matical work. Subsequently, the problem was discussed by Cardan and his contempo rary Tartaglia (Nicola Fontana) (1499

1557). All these men, however, arrived at

incorrect conclusions to the problem. About one hundred years later, in 1654, a French

man, Antoine Gombaud, the Chevalier de

M?r?, encountered the problem and, not pos

sessing the necessary calculating ability, sent it on to the mathematical prodigy of the

day, Blaise Pascal. Pascal (1623-1662) was born in Au

vergne, France, and at an early age showed

exceptional mathematical ability. When

only twelve, he discovered many of the the

orems of Euclidean plane geometry totally on his own. At sixteen he discovered the

beautifully rich "mystic hexagram" of pro

jective geometry, and he later invented and

constructed the first adding machine. This

astonishing and precocious activity came to

a complete halt in 1650 when, suffering from

fragile health, he abandoned mathematics

to devote himself to religious meditation and

philosophy. In fact, he is best known to the

lay person as the author of the first prose classics in modern French, the famous

Pens?es (1952) and Provincial Letters (1952). Three years later, in 1654, he returned

briefly to mathematics and made many dis

coveries relating to binomial coefficients, which he presented systematically in his

Trait? de Triangle Arithm?tique (see Pascal

[1665]). In this period he also became in

volved with the problem of points. Although Pascal made many other important contri

butions to mathematics, he has also been

called the "Greatest Might-Have-Been" in

the history of mathematics. Possessing such

extraordinary mathematical talents and

such keen intuition, he should have been

able to produce many more discoveries. Un

fortunately, during much of his life he suf

fered from the pains of acute neuralgia and

from the mental torments of religious fanat

icism. He died in 1662 at the age of thirty nine.

In contrast to Pascal's short, physically uncomfortable, and spasmodically produc tive life, the life of Pierre de Ferm?t (1601 1665) was enjoyable, peaceful, moderately

long, and continuously productive mathe

matically. He was born near Toulouse, be

came a lawyer, and devoted much of his

leisure time to the study of mathematics. He

evidently had no particular mathematical

training and evidenced no interest in math

ematics until he was past thirty, yet no

mathematician of his day made greater dis

coveries or contributed more to the field. He

invented, with Descartes, analytical geome

try; laid the technical foundations for calcu

lus; and founded the modern theory of num

bers, making it a full-fledged area of

abstract mathematics and the area in which

he was most prolific. A modest man, he pub lished little but was in constant correspon dence with many of the leading mathemati

cians of his day and had considerable

influence on his contemporaries. Ferm?t en

riched so many areas of mathematics with so

many important contributions that al

though he is sometimes called the "Prince of

November 1991 625



Amateurs," he has also been designated the

greatest French mathematician of the sev enteenth century. We shall soon see where Ferm?t fits into our story.

The Chevalier de M?r?, an able and ex

perienced gambler as well as a soldier, lin

guist, and classical scholar, made a rather

precarious living at dice and cards, making wagers according to his own system of prob abilities. Although he was not an accom

plished mathematician, he knew enough about the subject to pose some interesting problems, one of which was the problem of

points, for which his calculations did not

agree with his observations. He had pre sented the problem to Pascal, who became

quite interested in it and communicated it to Ferm?t. A remarkable correspondence fol lowed between these two great French

mathematicians (almost all of which is in tact and appears in Smith [1959]), in which the problem was correctly but differently solved by each man. In Pascal's letter to Ferm?t on 29 July 1654, he refers to the problem and possible solutions: "Your method is very sound and it is the first one that came to my mind in these researches, but because the trouble of these combina tions was excessive, I found an abridgment and indeed another method that is much shorter and more neat" (Smith 1959, 548).

His letter of 27 October 1654 to Ferm?t sug gests that the correspondence was spirited when they did not always agree with one another: "Monsieur. Your last letter satis fied me perfectly. I admire your method for the problem of the points, all the more be cause I understand it well. It is entirely yours, it has nothing in common with mine, and it reaches the same end easily. Now our

harmony has begun again" (Smith 1959, 564). In this correspondence of 1654, Pascal and Ferm?t jointly laid the foundations for the theory of mathematical probability, an event that Howard Eves calls a "great mo ment in mathematics" (1983, 8).

We noted that Pascal's life focused alter

nately on religion and philosophy and on mathematics. Interestingly, he joined the two using probability. Pascal argued in his

Pens?es (1952): "We know neither the exis

tence nor the nature of God .... Let us

weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If

you gain, you gain all; if you lose, you lose

nothing. Wager, then, without hesitation that He is" (Pascal 1952, 214-15).

Now let us examine the problem of points a bit more deeply. Suppose that de M?r? and another of his friends were playing a

seventeenth-century dice game. Each player bets thirty pistoles that his chosen number will turn up three times on a die before the other player's number comes up three times. After the game has been underway for a

while, de M?r?'s number, 5, has turned up twice while his opponent's 3 has turned up only once. At this point, de M?r? receives an

urgent message that calls him away, so the

game must stop. How should the players split the sixty pistoles on the table? De

M?r?'s friend contends that since his chances of getting two lucky throws are half as good as de M?r?'s chances of getting one

lucky throw, he should receive half as much of the pot as de M?r?: twenty pistoles to de

M?r?'s forty. De M?r?, however, argues that on the next throw of the die the worst that could happen to him would be to lose his

advantage, in which case the game would be even and he would be entitled to an even

split of thirty pistoles. However, if his next throw were a 5, he would win the original wager and pick up all sixty pistoles. De M?r?

contends, therefore, that even before the throw he is entitled to the thirty pistoles he is sure of plus fifteen more that he is half sure of; hence, he should receive forty-five to his opponent's fifteen. And he is right; Pas cal and Ferm?t decided this outcome early in their celebrated correspondence. They also considered other problems related to the

problem of points, such as the division of stakes when the two players are unevenly skilled or when more than two are playing.

With their work, the mathematical theory of

probability was well launched. In 1657, the great Dutch physicist Chris

tian Huygens (1629-1695) wrote the first formal treatise on probability, De Ratioci niis in Ludo Aleae (On reasoning in games of

chanceXsee Huygens [1801]), based on the




Ferm?t-Pascal correspondence. This was the best account of the subject until 1713, when Ars Conjectandi (The art of conjecturing) by Jakob (Jacques) Bernoulli (1654-1705) was

published posthumously. This was the first

published book devoted entirely to the the ory of probability.

Statistics

We mentioned earlier that probability had twin roots in two distinct lines of investiga tion. We have explored the solution of gam

bling problems at some length. Next let us

consider the processing of statistical data. Statistics has been defined as the science and art of gathering, analyzing, and making in ferences from data. Accurate and systematic counting of economic wealth, population, and even the plunder of war goes back to

antiquity. For example, the census in Israel,

Emperor Augustus's accounting sheets on

the Roman Empire, and William the Con

queror's inventory of his newly won English possessions recorded in the Domesday Book

(c. 1085) are but a few of the historic data

gathering activities. Significant statistical

investigations began, however, only when

merchants, particularly those representing insurance companies, needed probabilistic estimations of events.

The first statistics dealt with mortality.

Insurance appears to have been created as early as Roman times to protect merchant

sailing vessels. The first marine insurance

companies were established in the four teenth century in Holland and Italy; by the sixteenth century the idea had moved to

other countries. Although a life-insurance

policy had been underwritten by a small

group of men in London as early as 1583, a

true insurance company for this purpose was

not established until 1688. To put these op erations on a firm actuarial footing, how

ever, some mathematical determination of

probabilities was called for. John Graunt

(1620-1674), a London merchant, was the

first person to draw statistical inferences from analyses of mass data. In 1662 he pro duced a work that launched the discipline we now call mathematical statistics: Natural and Political Observations Made upon the Bills of Mortality (see Graunt [1676]). Indeed we sometimes call Graunt the "Father of Statistics." The bills of mortality from which Graunt drew his conclusions were originally yearly and weekly reports of the number of burials in various London church parishes. They seem to have arisen as early as 1532 to

keep track of the progress of the plague in London. Graunt's work included many con

clusions of varying validity and generality. He made these observations, among others: more male births occurred than female

births, women tended to live longer than

men, and the number of persons dying (ex

cept during epidemics) was fairly constant from year to year. On the lighter side, noting that physicians claimed they had half as

many male as female patients, he concluded that either physicians usually cured wom

en's infirmities or a larger proportion of men

died from their vices without resorting to medical aid! As a result of these researches, Graunt became one of the charter fellows of the Royal Society of London at its founding in 1662; he was the only shopkeeper or

tradesperson so elected. After Graunt's work with mortality ta

bles, the next important work was done by Oxford mathematician Edmund Halley (1656?1742) of comet fame. It was contained in his 1693 memoir, Degrees of Mortality of

Mankind, in which he made a careful study of annuities. After these pioneering efforts, the subject was carried forward by such peo

ple as Abraham Demoivre (1667-1754), a

French Huguenot who had emigrated to En

gland, where he came to know both Newton

and Halley. He wrote Doctrine of Chances in

1718 (see Demoivre [1756]), a famous publi cation that further developed the mathemat

ics of permutations and combinations and was the first English treatise on the subject, and Annuities upon Lives (1725), which played an important role in the development of actuarial mathematics and its relation to

life insurance. Demoivre later prepared a

November 1991 627



paper in which he described the normal

probability curve.

Following Demoivre, probability and sta tistics entered a transitional period, where the concepts were examined, applied to old and new problems beyond those of gambling and mortality tables from which it had its

beginnings, and simplified for general un

derstanding. By this time mathematicians were beginning to realize that many con

cepts from probability could not be sepa rated from statistics, for statisticians must consider probabilistic models to infer prop erties from observed data. Daniel Bernoulli

(1700-1782), one of the many members of the remarkable Bernoulli family, which is to

mathematics what the Bach family is to mu

sic, investigated the famous Petersburg par adox, first proposed by his older brother Nikolaus in 1713. The problem arose from a

game of chance in which one player tosses a

coin and a second player agrees to pay a sum

of money if heads comes up on the first toss, double the money if heads appears on the second toss, four times as much if on the third throw, eight times as much if on the fourth toss, and so on. The paradox arose

concerning how much should be paid in be fore the game to make it fair to both players; a wide discrepancy appeared between ordi

nary common sense and mathematical rea

soning, which said that an infinite sum was

necessary. Bernoulli also showed how calcu lus (which had been invented about sixty years earlier) could be applied to probabil ity. Leonhard Euler (1707-1783) system atized and organized the many probability problems, and Joseph Lagrange (1736 1813) also advanced the theory of probabil ity by applying differential calculus to it.

The theory necessary to treat problems in which the number of possible outcomes becomes infinite (the theory of continuous

probability) was developed by the French aristocrat, professor, politician, and mathe

matician, Pierre Simon de Laplace (1749 1827). Coming from very humble origins, he

eventually became a teacher at the Ecole Militaire in Paris, where he taught Napo leon Bonaparte, who later got him involved in the mercurial politics of France. Laplace

also helped to organize the world-famous Ecole Polytechnique and Ecole Normale and

became recognized throughout Europe as a

fine scientist and mathematician. Laplace's interests really were in the heavens; his

monumental five-volume work, Celestial

Mechanics, was published in 1799 (see La

place [1966]). But he also used the theory of

probability to obtain a statistical measure of

reliability of numerical results derived from data and to determine the likelihood that certain astronomical phenomena were due to definite causes rather than pure chance. He did more than anyone else to advance the

theory of probability by publishing in 1812 his Th?orie Analytique des Probabilit?s (An

alytical theory of probability), which orga nized and brought together all that was

known about probability and statistical the

ory. Unfortunately, he devoted little time to

explaining the steps in his analysis or in

polishing his work. He often would avoid full discussion by saying, "It is easy to see," if he knew or felt the results were correct. The American astronomer Nathaniel Bowditch

(1733-1838), when translating the Celestial Mechanics into English, remarked, "I never came across one of LaPlace's Thus it plainly appears without feeling sure that I had hours of hard work before me to fill up the chasm and find out and show how it plainly appears" (Burton 1985, 452-53). Neverthe

less, Laplace is often called the "Father of Modern Probability Theory" because he stands at the zenith of its development. He

understood, perhaps better than anyone else of his time, the significance of probability to the world. In his own words (Moritz 1958, 342), The most important questions of life are, for the most

part, really only problems of probability. Strictly speak ing, one may say that nearly all our knowledge is

problematical; and in the small number of things which we are able to know with certainty, even in the math ematical sciences themselves, induction and analogy, the principal means for discovering truth, are based on

probabilities, so that the entire system of human

knowledge is connected with this theory .... It is remarkable that probability, which began with the con sideration of games of chance, should have become the

most important object of human knowledge.

In the years following Laplace's contribu

tions, the concepts of probability continued




to be refined where necessary and applied to

many topics, including the burgeoning area

called statistics. Using probability theory, a

number of mathematicians derived various

types of mathematical distributions that

would describe various populations, includ

ing the Bernoulli and the Poisson distribu

tions, neither of which fits the normal dis tribution described by Karl Friedrich Gauss

(1777-1855), who devoted special attention to the normal curve, its equation, and its

applications. The beginning of statistical

analysis of census data was accomplished in 1829 by Lambert Adolphe Jacques Quetelet (1796-1874) in Belgium. Father Gregor

Mendel (1822-1884) related probability to genetics and hybridization in 1865. Francis Galton (1822-1911) discovered the law of regression and the correlation coefficient in

1877. Beginning in 1894, Karl Pearson

(1857-1936) applied probability to biology and created the area of study we now call biometrics. The Russian Andrey Markov

(1856-1922) developed his chain theory of probabilities; and Norbert Wiener (1894

1964), longtime professor at the Massachu setts Institute of Technology, expressing his belief that probability is the link between physics and mathematics, created

cybernetics.

Curricular Considerations

As we conclude this broad look at the devel

opment of probability and statistics, let us return to Laplace for a final comment. He

said in his Analytical Theory of Probability (Moritz 1958, 340): The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to

appreciate with exactness that which accurate minds

feel with a sort of instinct for which ofttimes they are

unable to account_It teaches us to avoid the illusions

which often mislead us; . . . there is no science more

worthy of our contemplations nor a more useful one for

admission to our system of public education.

One has only to glance through today's newspapers to see the extent to which the

language of probability and statistics has

become an integral part of our lives. Al

though he was writing about 175 years ago,

Laplace was right! Individuals need to know

statistical language to be able to understand

something as simple as weather reports and

forecasting, sports reports, and even adver

tising. Individuals also need a knowledge of

probability and statistics to function in daily society, to be able to read and understand

government and business documents, and to

avoid "doublespeak," the misleading use of statistics. Students need a knowledge of

probability and statistics to apply to the many situations they may confront in future

study in mathematics and other subjects. And it is important to appreciate aestheti

cally the beauty of the subject and its appli cations to technology, science, and nature. These comments were made in the first ar

ticle of the 1981 Yearbook of the National Council of Teachers of Mathematics

(Pereira-Mendoza and Swift 1981). One

hopes that no mathematics teacher would

seriously question these statements today, nor does any real disagreement remain with

Laplace's feeling that the study of probabil ity and statistics belongs in the school math ematics curriculum.

Returning to the NCTM's Curriculum and Evaluation Standards (1989), it is inter

esting to note that the Pascal-Fermat "prob lem of points" is explored on pages 138-39 as an example of mathematical model build

ing. And later on pages 167-75, standards 10 and 11 address very explicitly the content areas of statistics and probability recom

mended for inclusion in the high school mathematics curriculum. In statistics, six areas of knowledge are outlined so that all

students, and especially the college bound, will be able to transform data to aid in in

terpretation and prediction and to test

hypotheses using appropriate statistical tools. In probability five areas of content are

outlined that enable college-bound students to apply the concept of random variable to

generate and interpret a variety of different

probability distributions. The recommenda

tions are clear; for the benefit of all our

students, we must give increased attention to probability and statistics throughout the entire K-12 school program.

As with so many curricular recommenda

tions, it may be easy, for a while at least, to

ignore them. We mathematics educators

November 1991 629



have been doing so successfully with respect to inclusion of probability and statistics in

the secondary school mathematics curricu

lum for about one hundred years (or 175

years if we go back to Laplace's recommen

dation!). Curricular change is always a slow

process, but should something so important (and with which probably most people gen

erally agree) take so long? Is it not time to

force the issue and give all our students

these important concepts and skills, which

they will need to function in the twenty-first

century? Should we wait any longer? Can we

wait any longer?

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