WHICH DEPEND UPON THE PROBABILITY OF EVENTS

CHAPTER XV.

150

A

PHILOSOPHICAL

ON

PROBABILITIES.

titions; and_dividingitby^this

number

the _quotientor the

mean

benefit ofeach event is the mathematical hope ^itselforthe advantage^{relative tothe event. It}

isthe

same

with a losswhich becomes certain inthe longrun, howeversmall the disadvantage^oftheevent

may

^be.

This theorem upon^benefitsandlosses is analogous to thosewhich

we

have _alreadygivenupon ^theratios which are indicated by ^{the indefinite} repetition of events simple or

compound;

^{and, like them, it}proves that regularityends byestablishing ^itself even in the things which are most subordinated to thatwhich

we name

hazard.

When

the events arein greatnumber, analysisgives another very simple expression of the probability that thebenefit willbecomprised ^{withindetermined limits.}

This is the expression which enters again ^{into the} general law of probability given above in speaking of the probabilities which result from the indefinite multiplication of events.

The

_stability ofinstitutions which are based upon probabilities depends upon ^{the truth of} the preceding theorem. Butin orderthatit

may

be applied^tothem

it is necessary^that those institutions should multiply these advantageous ^{events for the sake of} numerous things.

There have been based upon ^the probabilities of

human

lifediversinstitutions, suchaslife annuitiesand tontines.

The

most _general and the most _simple method ofcalculating thebenefits andthe expenses^of these institutions- consists in reducing these ^{to actual} amounts.

The

annual interestof_unity is thatwhich

BASED 151 is called therateof^interest.

At

the endofeach year anamount_acquiresfor a factor unity plusthe rateof interest; ^it increases then according^to a geometrical progression ofwhichthisfactor is theratio.

Thus

in thecourse of timeitbecomes immense. _If,for

exam-ple,therateofinterestis--$orfivepercent,the_capital doubles very nearly ⁱⁿ fourteen _years, quadruples ⁱⁿ twenty-nine years, and in less than three centuriesit

becomes twomilliontimes_larger.

An

increase soprodigious hasgiven^{birth to} theidea ofmaking^{use ofitin orderto}pay^offthe public debt.

One

forms forthis purpose a sinking fund ^towhichis

devoted an annual fundemployed^{for the} redemption of public ^bills and without ceasing increased by ^the interest ofthebillsredeemed. Itis clear thatin the long run this fund will absorb a great part of the national debt. If,

when

the needsof the State

make

a loan necessary, apart of ^thisloanis devoted tothe increasingof the annual sinkingfund, the variation of public ^billswillbe _less; the confidenceofthe lenders and the probabilityof_retiring without loss of_capital loaned

when

one desires will beaugmented and ^will render theconditions oftheloan lessonerous. Favor-ableexperienceshave_fullyconfirmed these advantages.

But the fidelity in engagements and ^the stability, so necessary ^to the success of such institutions, can be guaranteed onlyby^a governmentⁱⁿwhichthe legisla-tive power ^{is divided}

among

^several independent powers.

The

confidencewhichthe necessary coopera-tionofthese powers inspires, doubles the strength^of the State, andthe sovereign himself gains thenⁱⁿlegal power more ^{than heloses in}arbitrarypower.

152

A

ESSAY

It results from that which _precedesthatthe actual capital equivalent ^to a

sum

whichis tobe paid only after a certain

number

_{of years} is equal ^{to this}

sum

multipliedby^the probability that^itwill be paid^atthat time and divided byunityaugmented by ^{the rate of} interestandraisedtoa powerexpressed by^the

number

of these_years.

Itiseasy^toapply^this principle to^lifeannuitiesupon one or several persons, andto savings banks, andto assurance societies of any^nature. Suppose ^that one proposes^to form a tableoflifeannuities according ^to a given^table of mortality.

A

^life annuity payable ^at the end offive years, ^for example, and^reducedtoan actual

amount

is, by^this principle,equal^tothe product of the two _following quantities, namely, ^the annuity divided by^{the fifth} power^of unityaugmented by^the rateofinterest andthe probabilityofpaying^{it. This} probability^is the inverse ratioof the

number

of indi-viduals inscribedinthetableopposite^totheage^ofthat one

who

settles the annuity ^to the

number

inscribed opposite^tothisageaugmented by^fiveyears.

Form-ing, then, aseries offractionswhose denominatorsare the products^ofthe

number

_{of persons} indicated inthe tableof mortalityas living atthe age ^ofthatone

who

settles the _annuity, by^{the successive}powersof unity augmented

by

^{therate ofinterest,} and whose numera-torsarethe products of the annuityby^the

number

of persons living atthe

same

age augmentedsuccessively by one year,

by

^two years, etc., the

sum

of these fractions willbe the

amount

_requiredforthelifeannuity atthatage.

Letussuppose ^that a person wishes by

means

of a

INSTITUTIONS 153 life annuity^to assure to his heirsan

amount

_payable

at theendofthe year of^hisdeath. Inorderto deter-mine thevalueofthis annuity, one

may

imagine^that the person borrows in life ata bank this capital and thathe_placesitatperpetual^interestin thesamebank.

It is clear that this same _capital will be due by^the bank tohis heirs attheendof the year of^hisdeath;

buthewill havepaid each year only the excess of the

lifeinterest overthe perpetual ^interest.

The

tableof life annuities will then

show

that which the person ought^to payannually^to thebank in order to assure thiscapital^afterhis death.

Maritime assurance, thatagainst^fireand_storms,and generally^allthe institutions ofthiskind, arecomputed on the same principles.

A

merchant having^vessels at sea wishes to assure their value and that of their cargoes against the dangers^thatthey

may

^run;inorder

to dothis,he_givesa

sum

toa

company

^whichbecomes responsible ^to him for the estimated value of his cargoesand his vessels.

The

ratioofthisvaluetothe

sum

which ought^tobe given^forthe_{price of}the assur-ance depends upon ^the dangers ^to which thevessels areexposed andcan be appreciated only

by

^numerous observationsupon ^{the fateofvessels}which havesailed from port^forthe

same

destination.

If the persons assuredshould give^totheassurance

company

only the

sum

indicated by ^{the calculus of} probabilities, ^this

company

^{wouldnotbe able to} pro-videforthe expenses of ^its institution_; itis necessary thenthatthey shouldpay^a

sum much

greater than the costofsuchinsurance.

What

thenistheiradvantage^?

Itis herethatthe consideration of themoral

disadvan-154

A

tage attached to an uncertainty becomes _necessary.

One

conceives thatthe fairest

game

becomes, ^as has alreadybeen _seen, disadvantageous, because the player exchanges ^{a certain stake for} an uncertain _benefit;

assurance

by

which one exchanges the uncertain for the certain ought ^to be advantageous. ^{It is} indeed thiswhich results from the rulewhich

we

have _given above for determining moral hope and by which one seesmoreover

how

far the sacrifice

may

extend which ought ^{to be}

made

to the assurance

company

by reserving always ^amoral _advantage. This

company

can then in procuring ^this advantage ^itself

make

a great benefit, ^ifthe

number

of the assured persons^is very large, a condition.necessary ^{to its} continued existence.

Then

its benefits

become

certainandthe mathematical and moral hopes coincide; ^for analysis leads to this general theorem, namely, ^{that if} the expectations^areverynumerousthetwo hopes approach each other without ceasing and end

by

coinciding ⁱⁿ the case ofan infinitenumber.

We

havesaidinspeaking^ofmathematical and moral hopes ^{that thereis a}moral advantage ⁱⁿ distributing the risksofabenefitwhich one expects over^{several of}

itsparts.

Thus

inorder tosend a

sum

of

money

^toa

distant part ^{it is}

much

better to send it on several vessels than to expose ^iton one. This one does

by means

of mutual assurances. If two _persons, each having the same

sum

_{upon two}differentvessels which havesailedfrom thesame _{port to}thesamedestination, agree ^{to divide} equally ^{all the}

money

^which

may

arrive, itis clear that by^this agreement^{each of}them dividesequallybetween thetwovesselsthe

sum

which

INSTITUTIONS BASED 155 he _expects. Indeed this kind of assurance always leavesuncertainty^{as to} the losswhich one

may

^fear.

But this uncertainty diminishes ⁱⁿ proportion ^as the

number

of policy-holders ^increases; the moral advan-tage ^increases more and

more

and endsbycoinciding with the mathematical advantage, ^{its natural limit.}

This renders theassociationofmutualassurances

when

itisvery numerous more advantageous^to the assured ones than the companies of assurance which, ⁱⁿ pro-portion to the benefit that they give, give a moral advantage always^{inferiortothe} mathematical advan-tage. Butthe surveillanceoftheiradministrationcan balance theadvantage^{of the} mutual assurances. All theseresults are, ashas alreadybeen_seen,independent ofthe lawwhichexpresses the moral_advantage.

One may

^look upon ^{a free} people ^as a great asso-ciationwhose

members

secure mutually^their proper-ties

by

supporting proportionally the charges of^this guaranty.

The

confederation ofseveralpeopleswould give^to themadvantages analogous^tothosewhicheach individual enjoysⁱⁿ the _society.

A

congress of ^their representativeswould discuss objects of a_utility

com-mon

toall and withoutdoubt~the system of weights, measures, and

moneys

proposed

by

^{the French}

sci-entists would be adopted ^{in this} congress ^as one of the thingsmost useful tocommericalrelations.

Among

^theinstitutionsfoundedupon^theprobabilities of

human

lifethe betterones are those in which, by meansofa _light sacrificeof his revenue, one assures his existence and that of his family^for a time

when

one ought ^{to fear to be unableto} satisfytheir needs.

As

faras

games

^are immoral, so^far theseinstitutions

156

A

areadvantageous^tocustoms byfavoring the strongest bents of our nature.

The

government ought ^{then to} encourage

them

and_respect

them

in thevicissitudes of public fortune^;since thehopes whichthey present look toward a distantfuture, they^areable to prosper only

when

shelteredfrom all inquietude during^their exist-ence. It is an advantage ^{that the} institution of a representativegovernment ^assuresthem.

Let us say a word aboutloans. It isclear thatin order toborrowperpetually^itisnecessary^topay^each year the product^ofthe _capital

by

^{the rate ofinterest.}

But one

may

^wishto discharge^this principal ⁱⁿequal payments

made

during a definite

number

of _years, payments which ^{are called annuities}and

whose

value

is obtainedin this manner.

Each

_annuityin orderto be reducedatthe actual

moment

_oughtto be divided

by

^apower ^ofunityaugmented

by

^{the rate ofinterest} equal^to the

number

_{of years} afterwhichthisannuity ought^to be_paid.

Forming

then a geometric progres-sion

whose

^first term is the annuity divided byunity augmented by^{therateofinterest,} and whose lastterm

isthis annuity divided by^the same _quantityraised to apowerequal^to the

number

of years duringwhichthe

payment

^shouldhave been made, ^the

sum

ofthis pro-gression ^will be equivalent ^to the _capital borrowed, which will determine the value of the annuity.

A

sinking fund ^isat bottom_{only a}

means

of converting into annuitiesa perpetual ^rentwith the soledifference that inthe caseof a loan by^{annuities the interest is} supposed^constant, whilethe interestoffunds acquired bythe sinking fund^{isvariable. Ifit}were the

same

in both cases, the annuity corresponding ^to the funds

INSTITUTIONS PROBABILITIES. 157 acquired would be formed by ^{these funds} and from this annuity theState contributesannually^tothe sink-ing^fund.

If one wishesto

make

alife loan itwillbe observed thatthetables oflife annuitiesgive thecapitalrequired to constitute a life annuity^at anyage, a simple pro-portion ^will give the rent which one ought^to pay^to the individual from

whom

the capital ^is borrowed.

From

these _principles allthe _possiblekinds of loans

may

^becalculated.

The

_principleswhich

we

have _just expounded con-cerning the ^benefits andthe lossesofinstitutions

may

serve to determine the

mean

resultofany

number

of observations alreadymade,

when

onewishes toregard the deviations ofthe results corresponding ^{to divers} observations. Letus designateby

x

the correction of the least result and

by x

_augmented successivelyby

g, q^', q"^{, etc., the}corrections of the following^results.

Let us

name

e, e', e", etc., the errorsof the observa-tionswhose lawof probability

we

willsuppose known.

Each

observation being a ^{function of}the _result, it is

easy^{to see that}bysupposing the ^correction

x

of this resulttobe verysmall, the erroreofthe first observa-tion

w

^rillbe equal^tothe product^ofx by^adetermined

coefficient. Likewise theerrore'ofthesecond obser-vationwillbe the product of the

sum

_{q plus x, by}a determinedcoefficient, and so on.

The

probability^of the errorebeing givenby^a

known

function, itwillbe expressed by^the same function ofthefirstofthe pre-cedingproducts.

The

probability of^e'willbe expressed by^the same function of thesecond of these _products, and soonofthe others.

The

probability ofthe

simul-158

A

ESSAY

ON

PROBABILITIES.

taneous existence of the errorse, f', c", etc., will be then proportional ^tothe product of these divers func-tions, a productwhichwill be afunction of x. This being granted, ^ifoneconceives a curvewhoseabscissa isx, and whosecorresponding ordinate^{is this}product, this curvewill represent the probability of the divers values of_x, whose limits will be determined by ^the limitsofthe errorse,e'', e",etc.

Now

letusdesignate by

X

the abscissawhich it is necessary ^to choose;

X

diminishedby

x

willbethe errorwhichwouldbe

com-mittediftheabscissa

x

werethetrue correction. This error, multiplied by ^the probability of

x

or by ^the corresponding ordinate of thecurve, willbethe product ofthe lossby^itsprobability,regarding, ^asone_should,

thiserror as alossattached to the choiceX. Multi-plying^this productby^thedifferentialof

x

the _integral taken from the first extremity of the curve ^to

X

^will

be the disadvantage ^of

X

_resulting from the valuesof

x

inferiortoX, For the values of

x

superior to X,

x

less

X

would be the errorof

X

^if

x

werethetrue cor-rection; the_integral of the product of

x

_bythe corre-sponding^{ordinate of the curve} and by^the differential of

x

will bethen the disadvantage^of

X

resulting from thevalues

x

_superior to x, ^this integral being taken from

x

_equal to

X

_up to the last extremity of the curve.

Adding

^this disadvantage ^to the preceding one, the

sum

willbe the disadvantage attached ^tothe choiceofX. This choice ought^tobe determinedby the condition that this disadvantage be a

minimum;

and a very simple calculation shows that for this,

X

ought ^{to be the abscissa} whose ordinate divides the curveinto two _{equal parts,} so thatitis thus probable

159 that the truevalue of

x

falls on neither the one side northe otherofX.

Celebrated geometricians have chosen for

X

the mostprobable value of

x

and consequently^thatwhich corresponds ^to the _largest ordinateof the _curve; but the preceding value appears^to

me

_evidentlythatwhich the theoryof probability indicates.

CHAPTER XVI.

CONCERNING ILLUSIONS IN THE ESTIMATION

No documento A PHILOSOPHICAL ESSAY (páginas 161-172)