A PHILOSOPHICAL ESSAY

A PHILOSOPHICAL ESSAY

ON

PROBABILITIES.

BY

PIERRE SIMON, MARQUIS ^DE LAPLACE.

TRANSLATED FROM THE SIXTH FRENCH EDITION

FREDERICK WILSON TRUSCOTT,

^PH.D. (HARV.), Professorof GermanicLanguages^intheU'est Virginia.University,

FREDERICK LINCOLN EMORY,

M.E. (WoR. POLY.INST.), Professorof Mechanicsand_AppliedMathematicsin the WestVirginia

University;Mem.Amer.Soc.Mtch. Eng.

FIRST EDITION.

FIRST THOUSAND.

NEW YORK:

JOHN WILEY &

SONS.

LONDON:

CHAPMAN

& HALL, ^LIMITED.

1902.

BY F.W.TRUSCOTT

F. L.EMORY.

ROBERT DRUMMOND PRINTER,NEW YORK

Annex

TABLE OF CONTENTS.

PART

I.

A PHILOSOPHICAL ESSAY ON PROBABILITIES.

CHAPTER

I.

PAGE

Introduction i

CHAPTER

II.

ConcerningProbability 3

CHAPTER

III.

GeneralPrinciples of theCalculusof Probabilities 11

CHAPTER

IV.

ConcerningHope ²⁰

CHAPTER

V.

AnalyticalMethodsof theCalculusof Probabilities 26

PART

II.

APPLICATION OF THE CALCULUS OF PROBABILITIES.

CHAPTER

VI.

GamesofChance ₅₃

CHAPTER

VII.

Concerning ^the Unknown Inequalitieswhich may ^Existamong

Chances_Supposedtobe Equal 56

iii

CHAPTER

VIII.

PACK Concerning^theLawsof Probabilitywhichresultfromthe Indefinite

Multiplication ofEvents 6

CHAPTER

IX.

Application of theCalculusof ProbabilitiestoNatural Philosophy.^. 73

CHAPTER

X.

Application^{of the}Calculus of Probabilities to theMoralSciences.. 107

CHAPTER

XL

Concerningthe Probability ofTestimonies 109

CHAPTER

XII.

Concerningthe SelectionsandDecisionsofAssemblies 126

CHAPTER

XIII.

Concerningthe Probability of theJudgments^ofTribunals 132

CHAPTER

XIV.

Concerning Tablesof Mortality,and theMeanDurations of_Life,

Marriage, andSomeAssociations 140

CHAPTER

XV.

Concerning^{the Benefitsof} Institutionswhich Depend upon ^the

Probability ofEvents 149

CHAPTER

XVI.

Concerning^{Illusionsinthe}Estimationof Probabilities 160

CHAPTER

XVII.

Concerning^theVariousMeansofApproaching Certainty 176

CHAPTER

XVIII.

HistoricalNoticeof theCalculusof Probabilities to1816 185

ERRATA.

Page 89, line 22, for ^{Pline read} Pliny

"

102, lines 14, 16, " _minutes "

days

"

143, line 25, "

sun soil

"

177, lines15, 17, 18,21, 22, 24,forprimaryread prime

"

182, line 5, for conjunctions read being binary

A PHILOSOPHICAL ESSAY ON

PROBABILITIES.

CHAPTER

I.

INTRODUCTION.

THIS

philosophical essay ^is the development ^of a lecture on probabilities which I delivered in 1795 ^to the normal schools whither Ihad been _called, by ^a decree of the national convention, ^as professor of mathematicswithLagrange. ^Ihave_{recentlypublished} upon ^the same _subjectaworkentitled The_Analytical Theory ofProbabilities. I present here without the aidof_analysisthe_principlesand_generalresultsofthis theory,applying themtothe mostimportant questions of_life,whichareindeedforthemost_partonlyproblems of probability. Strictlyspeaking ^it

may

^{even be said}

thatnearly^all ourknowledge^isproblematical; andin the small

number

_{of things} which

we

areableto

know

with _certainty, even in the mathematical sciences themselves, the principal

means

forascertaining ^truth induction and analogy ^arebased on probabilities;

A

PHILOSOPHICAL ESSAY

ON

sothatthe entiresystem^of

human

_knowledge is con- nected withthetheory^setforth in this essay. Doubt- lessitwillbe seen here withinterest thatinconsidering, even in the eternal principles of reason, justice, and humanity, only the ^favorable chances which are con- stantlyattachedtothem, there isa greatadvantage ⁱⁿ followingthese principlesand seriousinconveniencein departing from them: their chances, ^likethosefavor- able to _lotteries,alwaysend byprevailing ⁱⁿthemidst of the vacillationsof hazard. I hope ^{that thereflec-} tions given ^{in this} essay

may

^{merit the attention of}

philosophers and direct it to a _subject so worthy ^of engaging^theirminds.

CHAPTER

^II.

CONCERNING PROBABILITY.

ALL

_events, even thosewhich on account of their insignificance donotseem to followthe great laws of nature, arearesultofitjustas necessarily astherevolu- tions ofthesun. In ignorance of the^tieswhichunite such events totheentire system^ofthe universe, they have been

made

to dependupon^{final causes or} upon hazard, according^as they occur andare repeated with regularity, orappear without regard^toorder_;but these imaginary causes have _gradually receded with the widening bounds^of knowledge anddisappearentirely beforesound philosophy, which seesin

them

_{only the} expression of ourignorance of thetrue causes.

Present events are connected with preceding ones

by

^{atie based}upon ^theevident principle thata thing cannot occur without a causewhich _producesit. This axiom,

known

_bythe

name

ofthe principleofsufficient reason, extends even to actionswhich are considered indifferent

; thefreestwill is unable without a determi- native motive to give

them

birth; if

we

assume two positions with exactly ^similar circumstances and find that thewill is active inthe one and inactive in the

3

4

A

PHILOSOPHICAL ^BBS

AY

other,

we

say^thatitschoiceisaneffectwithout acause.

It is then, says Leibnitz, the blind chance of the Epicureans.

The

contrary opinion ^isan illusion ofthe mind, which, losing sightof the evasive reasons of the choice ofthe will in indifferent things, believes that choiceis determinedofitselfandwithout motives.

We

_ought ^{then to} regard the present ^state of the universe as the effect ofits anterior stateand as the cause of the one which is to follow. Given for one instantanintelligencewhichcouldcomprehend ^allthe forcesby which ^natureis animated andthe _respective situationofthebeings

who

_compose it anintelligence sufficiently vast to submit these data to analysis ^it would embraceinthe same formula the

movements

of the greatest bodies of the universe and those of the lightest atom; forit, nothing would be uncertain and thefuture, as the _past, would _{be present} to its eyes.

The human

mind_offers, inthe _perfectionwhich ithas beenable to give toastronomy,^{a feeble}ideaofthis in- telligence. ^Its discoveriesinmechanicsand_geometry, added tothat of universal gravity, have enabled it to comprehend ^{in the} same _analytical expressions the past and future states of the system ^of the world.

Applying^thesame methodto

some

other_{objects of}its

knowledge,^ithas succeededinreferring togeneral laws observed

phenomena

^{and in} foreseeing those which given circumstancesought^toproduce. Alltheseefforts inthesearchfortruthtend tolead it backcontinually to the vast intelligence which

we

have_{just mentioned,} butfromwhichitwillalways remaininfinitelyremoved.

This tendency, peculiar to the

human

race, is that which rendersitsuperior toanimals

;andtheirprogress

in thisrespect distinguishes nationsand_ages andcon- stitutestheirtrue glory.

Letus recall that formerly, andatno remote_epoch, anunusualrainoranextreme_{drought, a}comet having

in train a very long ^tail, the _eclipses, the aurora borealis, and in general ^all the unusual

phenomena

were _regarded as so

many

signs of celestial wrath.

Heaven

was invoked in order to avert their baneful influence.

No

_{one prayed}tohave the planetsandthe sun arrested in their courses: observation had soon

made

_apparent the _futility of such _prayers. But as these phenomena, occurring anddisappearing^at long intervals, seemedtooppose ^{theorderof nature, it}was supposed ^that Heaven, ^irritated by ^the crimes of the earth, had created them "to announce its vengeance.

Thus the long^tailofthe cometof1456spread ^terror through Europe, alreadythrown into consternation

by

the _rapid successes of the Turks,

who

had _just over- thrownthe

Lower

_Empire. Thisstarafterfour revolu- tions has excited

among

^usa very ^{different interest.}

The

_knowledge ofthelaws of the system^{of the}world acquired ⁱⁿ the interval had _dissipated the fears begotten bythe ignorance ^of the true relationship of

man

to the _universe; and _Halley, having recognized the identity of^thiscometwith those of the years 1531, 1607, and _1682, announced itsnextreturn forthe end of the year 1758 ^or the beginning of the year 1759.

The

learned world awaited with impatience^thisreturn which was to confirm one ofthe greatest discoveries that have been

made

in the _sciences, and fulfil the prediction of Seneca

when

he said, in speaking^ofthe revolutions ofthosestarswhich fall from an enormous

6

A

PHILOSOPHICAL ESSAY

ON

height:

"The

_daywill

come

_{when, by}studypursued through ^several ages, the things

now

concealed will appear with evidence; and_posteritywillbe astonished that truths so clear had _escapedus.^'

'

Clairaut then undertooktosubmitto analysis theperturbationswhich the comet had experienced by ^{theaction of the} two great planets, Jupiterand _Saturn; after

immense

cal- culations he fixed its next passage ^at the _perihelion towardthebeginningof April, 1759,whichwas_actually

verifiedbyobservation.

The

_regularitywhich astronomy shows us in the

movements

of the comets doubtless existsalsoinallphenomena. ^-

The

curve described bya simple molecule of ^airor vapor ^is regulated ⁱⁿa

manner

just as certain as the planetary^orbits; the only^difference between

them

^is

thatwhichcomesfrom our ignorance.

Probability^is relative, in part to ^this ignorance, ⁱⁿ part to our knowledge.

We know

thatofthree or a greater numberofevents a_single one ought^tooccur;

but nothing inducesus to believe thatoneof

them

will occurratherthan theothers. Inthisstateofindecision

itis impossible^forus toannouncetheiroccurrence with certainty. It is, however, probable ^thatone of these events, chosen at_will, will not occur because

we

see several casesequallypossiblewhich excludeitsoccur- rence, while only asingleonefavors ^it.

The

theory of chance consists in reducing ^all the events ofthe same kind to acertain

number

ofcases equallypossible, that isto say, tosuchas

we may

^be

equally undecided about in regard ^{to their} existence, and in determining the

number

of cases favorable to the eventwhose probability ^is sought.

The

ratio of

this

number

to that of all the cases _possible is the measure of this probability, which is thus simply a fraction whose numerator is the

number

offavorable casesandwhose denominatoris the

number

ofallthe cases possible.

The

preceding notion of probabilitysupposes that, in increasingⁱⁿ thesame ratiothe

number

of favorable casesandthatofallthe cases _possible, theprobability remains the same. In ordertoconvince ourselveslet us take two _urns,

A

^and B, the first containing ^four whiteandtwoblack _balls, and the second _containing only two white balls and one black one.

We _may

imagine thetwo blackballsof the firsturn attachedby a thread which breaks at the

moment when

one of them is seized in ordertobe drawn _out, andthefour whiteballsthus formingtwosimilarsystems. All the chanceswhich willfavorthe seizureofone oftheballs of the black system^{will lead toa black ball. If}

we

conceive

now

thatthe threadswhichunitethe ballsdo notbreak at_all, itisclearthatthe

number

of_possible chances will not change any more ^{than that of the} chances favorableto theextractionof the blackballs;

buttwoballs will be drawnfrom the urn atthe same time; the probabilityof drawing^{ablack ballfrom the} urn

A

will thenbe the sameas at first. Butthen

we

have obviously the case of urn

B

with the _singlediffer- ence that the three balls of this last urn would be replacedby^threesystems oftwo balls invariablycon- nected.

When

all the cases are favorable to an event the probability changes ^to certainty and its expression becomes_{equalto unity.}

Upon

^thiscondition, certainty

A

PHILOSOPHICAL ESSAY

and probabilityarecomparable, although^there

may

^be

an ^essential difference between the two states of the mind

when

^atruthis rigorouslydemonstrated toit, cr

when

itstillperceivesa small source oferror.

In thingswhich are only probable the difference of thedata,whicheach

man

hasinregard^tothem,^isone ofthe_principalcausesofthe diversity ofopinionswhich prevailⁱⁿregard ^tothe same_objects. Let ussuppose, forexample, ^that

we

havethree urns, A, B, C, one of which contains only black ^ballswhile thetwo others contain only white ^balls; a ballis to be drawn from the urn

C

and the probability^is

demanded

that this ballwill be black. If

we

do not

know

which ofthe threeurnscontainsblackballs only,sothatthereisno reasontobelieve thatitis

C

ratherthan

B

orA, ^these threehypotheses^willappear equallypossible,andsince a blackball can bedrawn_onlyinthe firsthypothesis, the probability ofdrawing^{it is}equal^to onethird. If itis

known

that the urn

A

containswhite balls only, the indecisionthen extends only^tothe urns

B

and_C, andthe probability thattheball drawnfrom the urn

C

will be black is one half. Finally ^this probability changesto certainty^if

we

are assuredthat the urns

A

and

B

containwhiteballs only.

It is thus that an _incident related to a numerous assembly^{finds various}degrees ^of credence, according to the extent of knowledge ^{of the} auditors. If the

man who

_reportsitis fullyconvinced of it and _if, by

hispositionand_character,he_{inspiresgreat}confidence, his statement, however extraordinary ^it

may

be, ^will have for the auditors

who

lack information the

same

degree of probability as an ordinary statement

made

by^thesame man, andthey^will have entire faithinit.

Butif

some

one of them

knows

that the sameincident is rejected by ^other equally trustworthymen, hewill be in doubt andthe incidentwillbediscredited by^the enlightened auditors,

who

will reject ^it whether itbe inregard ^{tofacts well} averred orthe immutable laws of nature.

It is tothe influence of the opinion ^ofthose

whom

themultitude judges^bestinformed andto

whom

ithas been accustomed to give ^its confidence in regard ^to themost important matters of^lifethatthe propagation ofthoseerrors is due whichintimes of ignorancehave covered the face of the earth. Magic and astrology offerus twogreat examples. These errors inculcated in infancy, adopted without examination, and having

for a basis only ^universal credence, have maintained themselves during a very long time^; but at last the progress of^sciencehas destroyed them inthe mindsof enlightened men, whose _opinion consequently has caused

them

to disappear even

among

^the

common

people, through the powerof imitationandhabit

w

^rhich had so generally spread

them

abroad. This power, the richestresourceofthe moral_world,establishesand conserves ina whole nationideas entirelycontrary^to those which it upholds elsewhere with the same authority.

What

_{indulgence ought}

we

not then to have foropinions^different from ours,

when

this differ- enceoftendependsonlyupon^thevarious pointsofview where circumstanceshave _placedus! Letus enlighten those

whom we

_judgeinsufficiently instructed; butfirst let us examine_criticallyour

own

_{opinions and weigh} withimpartiality ^theirrespective probabilities.

A

PHILOSOPHICAL ESSAY

The

difference of opinions depends, however, upon the manner in which the influence of

known

data is

determined.

The

_{theory of}probabilitiesholdstocon- siderationsso delicate thatitisnotsurprising thatwith the same datatwo _persons arrive at different _results, especially ⁱⁿ very complicated questions. Let us examine

now

thegeneralprinciples of^thistheory.

CHAPTER

III.

THE GENERAL PRINCIPLES OF THE CALCULUS OF

PROBABILITIES.

First_Principle.

The

firstof these _principles isthe definition itselfofprobability,which,^ashas been _seen,

isthe ratioof the

number

of favorable cases to thatof allthe cases_possible.

Second _Principle. But that supposes the various cases equally possible. ^If they ^are not so,

we

will determine first their respective possibilities, whose exact appreciation^isone ofthe mostdelicate points of the theory of chance.

Then

the probability will be the

sum

of the possibilities of each favorable case.

Letusillustrate thisprincipleby an example.

Let us suppose ^that

we

throwinto the air a _large and _very thin coin whose two _{large opposite faces,} which

we

will callheadsand_tails, are perfectly similar.

Let us find the probabilityof throwing headsat least one time intwo throws. It is clearthat fourequally possible cases

may

^arise, namely, heads at the first

andatthe secondthrow; headsatthe firstthrowand

tails at the second; ^{tails at}the firstthrow andheads at the second; finally, tailsatboth throws.

The

first

12

A

PHILOSOPHICAL ESSAY

three cases are favorable tothe eventwhoseprobability issought; consequently^this probability^is equal ^to|;

so that it isa bet ofthree to onethat headswill be thrownatleastonceintwothrows.

We

can countatthis

game

only threedifferent cases, namely, heads^atthefirstthrow, which dispenses with throwing a second time; ^{tails at} the first throw and heads^atthesecond; finally, tailsatthe firstandatthe second throw. Thiswould reduce the probability to

| ^if

we

should consider with d'Alembert these three cases asequally possible. But itis apparent^that the probabilityof throwing heads ^at the first throwis f^, while that of thetwo other cases is J, the first case being a simple eventwhich corresponds ^totwoevents combined^: headsatthefirstandatthe second throw, and heads at the firstthrow, tails at the second. If

we

then, conforming ^tothe second _principle, addthe possibilityf ^ofheads at the first throwto the _possi- bility J ^{of tails at} the first throw and heads at the second,

we

shall have _f for the probability sought, which _agrees with what is found in the supposition

when we

_playthetwo throws. This'suppositiondoes not change^{at allthe} chanceof thatone

who

bets on

this event; ^itsimply^{serves to}reduce thevarious cases tothe casesequally possible.

Third_Principle.

One

of themost _{importantpoints} ofthetheoryof probabilities and thatwhich lends the most to illusions is the

manner

inwhich these prob- abilitiesincrease or diminishby^{their mutual}combina-

tion. Iftheeventsareindependent ofoneanother, the probabilityoftheircombined existence is the product oftheirrespective probabilities.

Thus

the probability

13 of throwing one ^{ace with a} single die is ^; that of throwing two ^{aces in} throwingtwo dice at the same time is --$.

Each

face of theone_{being able} tocom- bine with the six facesofthe other, there are in fact thirty-six equally possible cases,

among

^{which one}

single case_givestwo aces. Generally the probability that a simple event in the same circumstances will occur consecutively a given

number

oftimesis equal^to the probabilityofthissimple event^{raised to}the power indicated by^thisnumber. Having^{thusthe successive} powers ^{of afractionlessthan}unitydiminishing without ceasing, an eventwhich depends upon ^aseriesof very great probabilities

may ^become

extremely improbable.

Suppose ^{then an incident} be transmitted to us by twenty ^{witnesses in such}

manner

that the first has transmitted it to the second, the second^tothe third,

andsoon. Suppose again^thatthe probability ofeach testimony be equal ^to the fraction T^{9 ;} that of the incident resulting from the testimonies will be less than .

We

cannotbetter compare^thisdiminution of the probability than with the extinction ofthe_lightof objects by^the interposition of several pieces of_glass.

A

_relatively small

number

_{of pieces} suffices to take

away

^{the viewofan} object that a _{single piece} allows us toperceive ⁱⁿadistinctmanner.

The

historiansdo not appear ^to have _paid sufficient attention to this degradation ^of the probability of events

when

seen acrossa great

number

ofsuccessivegenerations;

many

historical events reputed ^{as certain}would beat least doubtfuliftheyweresubmitted tothis test.

Inthepurely mathematical^sciencesthe mostdistant consequences participateⁱⁿthe certainty of the princi-

M A

PHILOSOPHICAL ESSAY

piefromwhich theyare derived. In the applications of analysis tophysics the^resultshave allthe_certainty of facts or experiences. But inthe moral sciences, whereeach inferenceis deduced fromthatwhich _pre- cedesitonlyⁱⁿa probable manner, however _probable thesedeductions

may

be, thechanceof error increases with theirnumber and _ultimatelysurpasses the chance of truth in the consequences very remote from the principle.

Fourth_Principle.

When

two events depend upon each _{other, the} probabilityof the

compound

^{event is}

theproduct^ofthe probabilityofthefirsteventandthe probability^{that, this}eventhavingoccurred, the second will occur. Thus in the preceding case of the three urnsA, B, C,ofwhich two contain only white ^balls and one contains only black balls, the probability of drawing a white ^ball fromtheurn

C

is f, since of the three urns onlytwo contain ballsof that color. But

when

a whiteballhas beendrawn fromthe urn C, the indecision relative to thatoneofthe urnswhichcontain only black ^balls extends only^to the urns

A

and B;

the probability ofdrawing^{a whiteball} from the urn

B

is _;the product^of\by , or _,isthen the probability ofdrawing two^{white balls at} one time from the urns

B

andC.

We

seeby^thisexample ^{theinfluence of}pastevents upon^the probabilityoffuture events. Forthe prob- abilityof drawing^{a white ball} from the urn _B, which primarily^is f,becomes _\

when

a white ball has been drawn fromtheurn

C

; itwould change ^to certainty^if ablackball hadbeen drawn from thesame urn.

We

will determine this influence by means ofthe follow-

ing principle, which is a _corollary of the preceding one.

Fifth Principle. ^If

we

calculate a_{priori the prob-}

abilityof the occurred eventandthe probability ofan eventcomposed ^ofthatone andasecond onewhichis

expected, the second probability divided by ^{the first}

will be the probability of the event expected, drawn from the observedevent.

Here is presented the question ^raised by

some

philosophers touching the ^influence of the _past upon the probability of the future. Let us suppose ^at the play^ofheads andtailsthatheads has occurred oftener than tails.

By

^{this alone}

^we

^{shall be led to believe}

that in the constitution of the coin there is a secret cause which favors it.

Thus

in the conduct of life

constant happiness ^is a proof of competency which should induce us to employpreferablyhappy persons.

But^ifby^the unreliability ofcircumstances

we

arecon- stantlybrought back^{toa state} of absoluteindecision,

if,forexample,

we

_changethe coinateachthrowatthe play ^ofheadsandtails, the_past can shedno_lightupon the futureanditwould beabsurdtotake account ofit.

Sixth _Principle.

Each

of the causes to which an observed event

may

^{beattributedis indicatedwith} just as

much

likelihood as thereisprobability that theevent will take _place, supposing the event ^to be constant.

The

probabilityof the existence of any one^{of these} causesisthen afractionwhose numerator isthe prob- abilityoftheevent_resultingfrom thiscause andwhose denominator is the

sum

of the similar probabilities relative to allthe causes; ^ifthesevarious causes, con- sideredapriori, areunequally probable,^itisnecessary,

16

A

PHILOSOPHICAL ESSAY

in placeofthe probabilityof the event_resultingfrom each cause, to employ^the product^ofthis probability bythe possibility of thecauseitself. This^isthe funda- mental_principleofthisbranchofthe_analysisofchances whichconsistsin passing from events^tocauses.

This _principle gives the reason

why ^we

^attribute

regulareventsto a particular cause.

Some

philosophers have thought ^{that these} eventsare lesspossible than others and that at the play of heads and tails, for example,^thecombinationinwhich headsoccurs twenty successive times is less easy^{in its} nature than those whereheadsandtailsaremixedinan_irregularmanner.

But this opinion supposes ^that past events have an influence on the possibility of future events, which is

notatalladmissible.

The

regular combinations occur more _rarelyonly because they^areless numerous. If

we

seek a causewherever

we

_{perceive symmetry,} itis

notthat

we

regard a symmetrical event^aslesspossible thanthe others, but, since^this event ought ^{to be the}

effect ofa_regular causeor that of chance, the ^firstof these suppositions ^is more probable than the second.

On

a table

we

see letters arranged ^{in this} order,

Constantinople,

^and

^we

judge^thatthisarrange- mentisnotthe resultof chance, not because itisless possible than the others, ^{for if this} word were not employed ⁱⁿ any language

we

should not suspect ^it

came

from anyparticular cause, butthisword_being in use

among

us, ^itis incomparably

more

_probable that some _person has thus arranged the ^{aforesaid letters} thanthatthisarrangement^isdue tochance.

This is the place to definethe word extraordinary.

We

arrangeⁱⁿour thought^allpossibleeventsinvarious

classes_; and

we

_regardas extraordinary those ^classes which include a very smallnumber.

Thus

atthe play ofheads and tails the occurrence ofheads a hundred successivetimes appears^tousextraordinary because^of thealmost infinite

number

of combinationswhich

may

occurina hundred_throws; andif

we

divide the

com-

binations into regular ^seriescontainingan _{order easy} tocomprehend, and into irregularseries, the latterare incomparably more numerous.

The

_drawing of a white ball from an urnwhich

among

^{amillion balls}

contains onlyoneofthiscolor,the others being black, would_appearto us likewise extraordinary, because

we

form only two classes of events relative to the two colors. But the drawing ^{of the}

number

_475813, for

example, ^from anurn thatcontains a millionnumbers seems to us an ordinary event; because, comparing individually the numbers with one another without dividing them into classes,

we

have no reason to believe that one of

them

will appear sooner than the others.

From

what _precedes,

we

_{oughtgenerally}toconclude thatthe more extraordinary the event, the greater the need of its being supported by strong proofs. For those

who

attest it, being able ^todeceive or to have been _deceived, these two causes are as

much more

probable^asthe_reality of the eventisless.

We

shall see this particularly

when we come

to speak of the probabilityof testimony.

Seventh_Principle.

The

probabilityofafutureevent

is the

sum

of the products of the probability of each cause, drawn from the event observed, by ^the prob- ability that, this cause _existing, the future eventwill

A

PHILOSOPHICAL ESSAY

occur.

The

_{following example} ^{will illustrate this}

principle.

Letusimagine an urnwhichcontains onlytwoballs, each ofwhich

may

^{be eitherwhite or black.}

One

of these ballsis drawn andis putback intothe urn before proceeding ^to a

new

draw. Suppose ^{that inthe first} two draws white balls have been drawn; the prob- abilityof again drawing^{awhite ballatthe third}draw

isrequired.

Only two hypotheses can be

made

here^: either one of the balls is whiteandthe other _{black, or}bothare white. In the first hypothesis the probability ofthe event observed is J; ^{it is} unity or _certainty in the second. Thus in regarding these hypotheses ^{as so}

many

causes,

we

shall have for the sixth principle

% and _| for their respective probabilities. Butifthe

first hypothesis occurs, the probability of drawing ^a whiteball atthethirddrawis^; itisequalto certainty in the second hypothesis; multiplying then the last probabilitiesbythose of the corresponding hypotheses, the

sum

oftheproducts, orT⁹^, will be the probability ofdrawing^{a whiteball atthethird draw.}

When

the probabilityof a _single eventis

unknown we may

suppose ^it equal ^to any^{value from zero to} unity.

The

probability ofeach of these hypotheses, drawnfrom the event observed, is, by ^the sixth prin- ciple, a fractionwhose numeratoris the probability of theeventin thishypothesisand whose denominator is

the

sum

of thesimilar probabilities relative to all the hypotheses.

Thus

the probability thatthepossibility of theeventiscomprised within given ^limitsisthe

sum

of the fractionscomprised^within these limits.

Now

if

we

_multiply each fraction by ^the probability of the future event,determined inthe corresponding hypothe-

sis,the

sum

ofthe products^{relativetoall}the hypotheses will be, bythe seventh _principle,theprobability ofthe future event drawn from the event observed.

Thus we

find thatan eventhaving^occurred successivelyany

number

of _times, the probability that ^{it will} happen again thenexttime is equal^{to this}

number

increased by unity divided by ^the

same

number, ^increased by twounits. Placing the mostancientepoch ^of history atfivethousand years ago, ^{or at} 182623 days, andthe sun having ^risen constantly ⁱⁿ the interval at each revolution of twenty-four hours, ^itis a betof1826214 to one that it will rise again to-morrow. But this

number

isincomparablygreater^forhim who,recogniz- ingⁱⁿthe totality of

phenomena

^theprincipalregulator of days and seasons, sees thatnothing ^atthe present

moment

canarrest the course ofit.

BuffoninhisPoliticalArithmeticcalculates differently the preceding probability.

He

_supposes thatitdiffers from unity onlyby^afractionwhose numeratoris unity and whose denominator is the

number

2raised to a powerequal^tothe

number

ofdayswhich have_elapsed since the epoch. But the true

manner

of relating pastevents with the probabilityofcausesand of future eventswas

unknown

to this illustriouswriter.

CHAPTER

IV.

CONCERNING HOPE.

THE

probability ofevents serves to determine the hope ^{or the fear of}persons ^{interested in their exist-} ence.

The

word _hope has various acceptations; ^it expresses generally the advantage ^{of that} one

who

expects a^{certain benefitin}suppositionswhich areonly probable. This advantage ⁱⁿ the theory ofchanceis

a product ^of the

sum

_hoped forby^the probabilityof obtainingit; ^itisthe _partial

sum

which ought^{to result}

when we

do not wish torun the risks ofthe eventin supposingthat the divisionis

made

proportional ^tothe probabilities. This divisionis the onlyequitable one

when

allstrange circumstances^areeliminated; because an equal degreeof probability givesan _{equal right to} the

sum

_hoped for.

We

will call this advantage mathematical_hope.

EighthPrinciple.

When

the advantagedepends on severaleventsit is obtained bytaking the

sum

ofthe products ^oftheprobabilityofeach event

by

^thebenefit attached toits occurrence.

Letus apply^this principle to

some

_examples. Let

us suppose ^{that at} the play ^of heads and tails Paul receivestwo francsifhethrows headsatthefirstthrow and five francs if he throws it only ^at the second.

Multiplyingtwofrancs by^the probability of the first case, andfive francs

by

^theprobability of the second case, the

sum

of the _products, or two and _{a quarter} francs, willbe Paul's advantage. ^{It is}the

sum

which he ought^to giveⁱⁿadvance tothatone

who

_{has given} him this advantage; for, in order to maintain the equalityof the _play, the throw ought ^{to be} equal ^to theadvantage which^itprocures.

IfPaulreceives two francs

by

throwingheadsatthe

firstandfive francsby throwing ^itatthesecond throw, whether hehasthrownit ornotatthe first, the prob- ability ofthrowing heads^atthe secondthrow_{being ,} multiplyingtwo francs andfive francsby ^the

sum

of these products ^will give threeand one half francsfor Paul's advantage andconsequently^{for his} stakeatthe game.

Ninth _Principle. In a series of probable events ^of which the onesproduce^{a benefit}andthe others a_loss,

we

shallhave the advantage which ^{results from it}

by

making^a

sum

of the products of the probabilityofeach favorable event by^{the benefit} which it procures, and subtracting from this

sum

that ofthe products ^of the probability ofeach unfavorable event

by

^{the loss which}

isattachedto it. Ifthe second

sum

isgreater than the

first,thebenefitbecomesa lossand hope ^is changed ^to

fear.

Consequently

we

ought always ^{inthe conduct oflife} to

make

the product ^ofthe benefit hoped for, by ^its probability,^atleastequal^tothesimilarproduct^relative

A

PHILOSOPHICAL ESSAY

to theloss. Butitisnecessary, ⁱⁿ orderto attain _this, to appreciate exactly the advantages, the losses, and their respective probabilities. Forthisa great accuracy of mind, a^delicate judgment,and a great experience in affairs is necessary; itis necessary^to

know how

to guard one's ^selfagainstprejudices, illusions offearor hope, and erroneous_ideas, ideasoffortune and_happi- ness, withwhichthe majority^ofpeople ^{feed theirself-} love.

The

application ofthe preceding principles to the following question has greatly exercised ^the geometri- cians. Paul plays^atheadsandtailswith the condition of_receivingtwo francsifhe throws headsatthe first thro\v, four francs ifhe throws it only^at the second throw, _eight francs if he throws it only^at the third,

and so on. His stakeatthe playought to be, accord- ing ^{to the} eighth principle, equal ^to the

number

of throws, so ^{that if} the

game

^{continues to} infinitythe stake ought ^{to be infinite.} However, no ^reasonable

man

wouldwish toriskat this

game

^{evena small}sum,

for example ^{five francs.}

Whence

comes this differ- encebetweenthe resultofcalculationandthe indication of

common

sense?

We

soon recognizethatitamounts tothis: thatthe moral advantage which ^{a benefit} pro- curesforusisnot proportional^{tothis benefit}andthat

it depends upon ^athousand circumstances, ^often very difficult to define, but ofwhichthe most _general and most_importantisthat of fortune.

Indeeditisapparent^thatonefranchas

much

_greater value forhim

who

_{possessesonly a}hundredthan for a millionaire.

We

_ought then to distinguish ⁱⁿ the hoped-for ^{benefit its} absolute from its relative value.

Butthe latteris regulatedby^the motiveswhich

make

it desirable, whereas the first is independent of them.

The

_{general principle} for appreciating ^{this relative} value cannot be _given, but here is one _proposed by Daniel Bernoulliwhichwill servein

many

^cases.

Tenth_Principle.

The

relativevalueof an_infinitely small

sum

isequal ^toits absolutevalue dividedby^the total benefit of the person interested. This supposes that everyone has a certain benefitwhosevalue can never beestimatedas zero. Indeedeventhatone

who

possesses nothing always gives ^to the product of^his labor andtohis hopes ^{a value at least} equal ^{to that} which isabsolutelynecessary ^tosustainhim.

If

we

_applyanalysis tothe principle justpropounded,

we

obtain the following^rule: Letusdesignatebyunity the part of the fortuneof an individual, independent ^of his expectations. ^If

we

determine the differentvalues thatthisfortune

may

have by^virtueof these expecta- tionsandtheir probabilities,the product of these values raised respectively to the powers ^indicated by ^their probabilities^will be the physical ^fortunewhich would procure ^fortheindividual the

same

moral advantage whichhereceivesfrom the part of his fortunetakenas unityand fromhis expectations; bysubtracting unity fromtheproduct, thedifference willbe the increase of the physical ^fortunedue toexpectations^:

we

will call thisincreasemoral_hope. Itiseasyto seethat itcoin- cideswith mathematical hope

when

the fortune taken as unity becomesinfinite in reference to the variations which it receives from the expectations. But

when

these variations are an appreciable partofthis unity

24

A

PHILOSOPHICAL ESSAY

thetwo hopes

may

^differvery materially

among

^them-

selves.

This rule conduces to results conformable to the indications of

common

sensewhich can by^this

means

be appreciated with

some

exactitude.

Thus

in the preceding question ^{it is} found that if the fortune of Paul istwo hundredfrancs, he oughtnot reasonably^to stake morethan ninefrancs.

The same

rule leads us again to distribute the danger ^{over several} partsof a benefitexpected^ratherthantoexposethe entire benefit to thisdanger. ^It results similarly that atthe fairest

game

^{the lossis} always greater than the gain. Let ussuppose, ^forexample,^thata playerhaving^afortune ofonehundredfrancsrisks_fiftyatthe play of headsand

tails; his fortune after his stake at the play ^will be reducedtoeighty-seven francs, thatisto say, ^{this last}

sum

would _procure for the player the

same

moral advantage ^{as the stateofhis fortuneafter the stake.}

The

_play is then disadvantageous even in the case where the stake is equal ^to the product of the

sum

hoped ^for,by ^its probability.

We

_{can judge by}this oftheimmorality^of

games

^{inwhich the}

sum

_hopedfor is below this product.

They

^subsist only by ^false reasoningsand by^the cupiditywhich_they excite and which, leading the people ^{to sacrificetheir} necessaries to chimerical hopes whose improbabilitythey ^arenot in conditionto appreciate, arethe source of an_infinity ofevils.

The

disadvantage of

games

of chance, theadvantage ofnot exposing^tothe same danger^the whole benefit thatisexpected, andallthesimilar results indicatedby

common

sense, subsist,whatever

may

^{be the function}

25 of the physical ^fortune which for each individual expresses ^his moral fortune. It is enough ^{that the} proportion ^of the increase of this function to the increase of the physical ^fortune diminishes in the measurethatthelatterincreases.

CHAPTER

V.

CONCERNING THE ANALYTICAL METHODS OF THE CALCULUS OF

PROBABILITIES.

THE

application of the_principlewhich

we

have_just expounded ^{to the various} questions of probability requires methods whose investigation has given ^birth to several methods of analysis and especially to the theory of combinations and to the calculus of finite differences.

If

we

form the product^ofthe binomials, unity plus thefirst letter, unity plusthe second _letter, unityplus the third _letter, and so on up ^{to n} letters, and sub- tract unity from this developed product, the result willbe the

sum

ofthe combination of^all these letters taken one by^one, two by two, three

by

^{three, etc.,} each combination having unity ^for a coefficient. In order to have the

number

ofcombinations ofthese n letterstakensby^s times,

we

shall observethatif

we

suppose^{these letters}equal

among

themselves, the_pre- ceding product ^will

become

the nth power ^{of the} binomialone _{plus the} firstletter; thus the

number

of combinations ofnletters takens by^{stimeswillbe the} coefficient ofthe sth power ^{of the first letter in the}

27 development ^{in this binomial;} and this

number

is

obtained by means^{of the}

known

binomial formula.

Attention must_{be paid} tothe _respective situations of theletters ineach combination, observing ^thatifa second letteris joined ^tothefirst it

may

^be placed ⁱⁿ the firstorsecond _position which _gives two combina-

tions. If

we

_{join to} thesecombinations a third letter,

we

_{can give}it ineach combination thefirst,the second, and the third rank which forms three combinations relative to each of thetwo_others, in all sixcombina-

tions.

From

this it is easy ^to conclude that the

number

ofarrangements^ofwhichs lettersare suscepti- bleisthe product of thenumbersfrom _{unity to}s. In orderto payregard ^to the _respective positions ofthe lettersitisnecessary then to multiplyby^this product the

number

of combinations of nletters sbys times, which is tantamountto taking

away

^the denominator ofthe coefficient of thebinomialwhich _expressesthis

number.

Let usimagine a lotterycomposed ^{of n}numbers, ^of which rare drawnat each draw.

The

probability^is

demanded

ofthe drawing ^{of s} given numbersin one draw.

To

arrive at this letus form a fractionwhose denominatorwill be the

number

ofall the cases possi- ble or ofthe combinations of n letters taken r

by

^r

times, andwhose numeratorwill be the

number

ofall the combinationswhich contain the given^s numbers.

This last

number

isevidently^{that of} the combinations of the othernumbers taken nless sby^{nless stimes.}

This fraction will be the required probability, and

we

shall easily^find that it can be reducedto a fraction whose numerator is the

number

ofcombinations of r

28

A

PHILOSOPHICAL ESSAY

numberstakensby^stimes,and whose denominatoris the number of combinations of n numbers taken similarly^sby^stimes.

Thus

in the_{lottery of}France, formed as is

known

of 90 numbers ^of whichfive are drawnateach draw, the probabilityofdrawinga given combination^is -&> or T

V

;the _lotteryought ^thenforthe equalityof the play^to give eighteen times the ^stake.

The

^total

number

of combinations two by two ^{of the} 90 numbers^is4005^, andthatof the combinations two by two ^{of 5} numbers is 10.

The

probability of the drawingof a given pair^is then

3-^-5-, andthe _lottery ought^togive^fourhundred andahalftimesthe stake;

itought^togive 11748 ^timesfora given tray, 511038 timesfora quaternary,and 43949268 ^timesforaquint.

The

_lottery^isfar from giving the player these advan- tages.

Suppose ^{in an urn}awhite _balls, bblack _balls, and

after havingdrawn aballit is put back intothe urn;

the probability^is asked thatin

number

of draws

m

whiteballs and n

m

blackballs will be drawn. It is clear that the

number

of cases that

may

^{occur at}

each drawing ^{is a}-j-b.

Each

case of the second drawingbeing^{able to}combinewithallthe cases of the

first, the

number

of possible cases in two drawings^is the square ^ofthebinomial a-\-b. In thedevelopment ofthissquare, the square^ofaexpresses the

number

of casesinwhichawhite ballistwicedrawn, ^{the double} product ^ofa by ^b expresses ^the

number

of cases in

whichawhiteballandablackballaredrawn. _Finally, thesquare of^bexpresses the

number

of casesin which two black balls are drawn. _{Continuingthus,}

we

see generally ^that the th power ^{of the binomial a}

+

^b