CONCERNING ILLUSIONS IN THE ESTIMATION OF PROBABILITIES

CHAPTER XVI.

CONCERNING ILLUSIONS IN THE ESTIMATION

appreciating withjusticethe inconveniences of theones andthe_others,andthe probabilityof the proper

means

toguard^ourselvesagainst them. ^Itisthiswhichleads alternately to despotism and to anarchy ^the people

who

aredriven from the stateof repose^towhich_they neverreturnexcept^afterlong andcruel agitations.

This vivid impression which

we

receive from the presence of events, and which allows us scarcely ^to remark the contrary events observed

by

others, ^is a principalcause of erroragainstwhich onecannot suffi-cientlyguard ^himself.

Itisprincipally at

games

^ofchancethata multitude ofillusionssupporthope and^sustainitagainst unfavor-able chances.

The

majority of those

who

_play at lotteries do not

know how many

^{chancesare to their}

advantage,

how many

^are contrary ^to them.

They

seeonly the possibility by a small stake of gaining a considerable sum, andthe projectswhichtheir imagi-nation brings forth, exaggerate ^{to their} eyes the probability ofobtaining it; the poor

man

especially, excited by^{the desire ofabetter}fate, risks atplay^his necessities byclinging to the most unfavorable

com-binations which _promise

him

a great ^benefit. All would be without doubt _surprised by ^the

immense number

ofstakeslost ifthey could

know

ofthem; but one takescare on the contrary^to give^to the winnings agreatpublicity,whichbecomes^a

new

cause of excite-mentforthisfunereal play.

When

anumberinthe _lotteryofFrancehas notbeen drawn fora long time the crowdis eager ^to cover it

with stakes.

They

judge ^since the

number

has not been drawn for a long time^thatit ought ^{at the} next

A

ESSAY

drawing ^{to be} drawn in preference ^{to others.} So

common

anerrorappears^to

me

to rest upon^anillusion by which one^iscarriedbackinvoluntarily to the_origin of events. It _is, for example, very improbable ^that atthe play ofheadsandtails onewillthrowheadsten timesin succession. This improbabilitywhich strikes us indeed

when

ithas happened ^nine times, leads us tobelieve that atthe tenth throwtailswill bethrown.

Butthe past indicatingⁱⁿ the coin a greater propensity for heads than for tailsrenders the firstoftheevents moreprobable than thesecond_; itincreases asonehas seen theprobabilityofthrowing heads^atthe following throw.

A

similarillusionpersuades

many

people^that one can _certainlywin ina _lotterybyplacing each time upon^the

same

_number,untilitisdrawn,^{a stake}whose product surpasses the

sum

ofallthe stakes. But even

when

similar speculationswouldnot often be _stopped by^theimpossibilityof_sustaining

them

_theywould not diminish themathematical disadvantage ofspeculators and _they would increase their moral disadvantage, since ateachdrawingtheywouldriska verylarge part oftheir fortune.

Ihaveseen men, ardently desirous ofhaving^{a son,}

who

^couldlearn only with anxiety^ofthebirthsofboys

in the

month when

_theyexpected^to

become

fathers.

Imagining^{thattheratioof thesebirths to those of}girls

ought^tobe the sameatthe endofeach month, they judged^thattheboys already bornwould render more probable the ^birthsnext of_girls.

Thus

the extraction of a white ball from an urnwhich contains alimited

number

ofwhiteballsandof blackballs increasesthe probabilityof extracting a black^{ball at the} following

163 drawing. But this ceases to take place 'when the

number

of balls in the urn isunlimited, as one must suppose ⁱⁿ order to compare ^{this case with that of}

births. _If, in the course of amonth, ^there wereborn

many

more boys ^than girls, one might suspect ^that toward the time of ^their conception a general cause hadfavored masculine conception, whichwouldrender more probable the^birth next of a boy.

The

_irregular events of nature are not exactly comparable ^{to the} drawing ^ofthe numbers ofa_lotteryinwhich all the numbersaremixedat eachdrawing^{insuch a}

manner

as to render the chances of their drawing perfectly equal.

The

frequency ofoneofthese eventsseemsto indicate a cause _slightly favoring ^it, which increases the probability ofits next return, and its repetition prolonged^fora longtime,suchas along^seriesof rainy days,

may

develop

unknown

causesforits change; ^so that at each expected ^event

we

are not, as at each drawing ^{of a} lottery, led back to the same state of indecisionin regard^towhat ought^to happen. Butin proportion ^as the observation of these events is

mul-tiplied, the comparison ^{of their results with those of}

lotteriesbecomes

more

exact.

By

^{an illusion} contrary^to the preceding ones one seeksin the_past drawings^ofthelotteryofFrancethe numbers mostoftendrawn, ⁱⁿorder to form combina-tions upon which one thinks to place the stake to advantage. But

when

the

manner

inwhichthemixing of the numbers in thislottery^is considered, the past ought^tohavenoinfluenceupon^{the future.}

The

_very

frequentdrawings of a

number

areonly the anomalies ofchance; ^Ihave submittedseveral ofthem to

calcula-164

A

PHILOSOPHICAL ESSAY

ON

tion and have constantly found^that they^areincluded within the limits which the supposition of an equal possibilityofthedrawing ^{of allthe} numbers allows us to admitwithoutimprobability.

Ina long^seriesofeventsofthesame^{kind the}_single chances of hazardought sometimes ^toofferthesingular veinsof good^{luck or} bad luck whichthe majority ^of playersdonotfailto attribute toa kind of_fatality. It

happens^oftenin

games

which depend ^atthesame time upon ^hazardand upon ^the competency^{of the} players, that thatone

who

loses, troubled by^{his loss, seeks to} repair^itby^hazardous throwswhich he would shunin anothersituation; thus he aggravates^his

own

ill luck and _prolongs its duration. It is then that prudence becomes_necessaryandthatitis of importance^to con-vince oneselfthat the moraldisadvantage attached^to unfavorable chancesisincreased by^{theill luckitself.}

The

_opinionthat

man

_{has long} been _placed inthe centre of theuniverse, considering himself thespecial objectofthe caresofnature, leads each individual to

make

himself the centre of a

more

or less extended sphere and to believe that hazard has preference ^for him. Sustainedby^this belief, players ^{often risk} con-siderable sums at

games when

they

know

that the chances are unfavorable. In the conduct of life a similaropinion

may

sometimes have _advantages; but mostoftenit leads to disastrous enterprises. Hereas everywhere ^{illusions are} dangerous and^{truth aloneis} generally^useful.

One

of the great advantages^ofthecalculusof prob-abilitiesistoteach us to distrustfirstopinions.

As we

recognize ^that they ^often deceive

when

_they

may

^be

165 submitted to calculus,

we

_ought to conclude that in other matters confidence should be given only ^after extremecircumspection. Letusprove^thisby example.

An

urn containsfourballs,blackandwhite,butwhich are notall ofthe

same

color.

One

of these ballshas been drawn whose color iswhiteand which has been putbackinthe urninordertoproceed again^{to similar} drawings.

One demands

theprobabilityof extracting only black^ballsinthefourfollowing drawings.

Ifthe white and blackwere in equal

number

this probabilitywouldbe thefourthpower^oftheprobability of extracting a black^{ball at} eachdrawing; itwould be then_T^. Butthe extractionofa white ball atthe

first drawing ^{indicates a}superiorityⁱⁿ the

number

of whiteballs in the urn; for ifone_supposes in theurn three white balls and one black the probability of extracting a white^{ball is}|; ^{itis if}one _supposestwo whiteballs and two _{black; finally}itisreduced toJ ^if one_supposesthreeblackballsand onewhite. Follow-ing the _principle of the probability of causes

drawn

from events theprobabilities ofthese threesuppositions are

among

^{themselves as the} quantities ^, f, ^; they are consequently equal ^to |, f, ^. It isthusa betof 5 against ^{i that}the

number

of blackballs is inferior, or at themostequal, to that of thewhite. It seems thenthatafterthe extractionofawhiteballatthe first

drawing, theprobability ofextractingsuccessively four black balls ought ^to be less thanin the case of the equalityof the colors or smaller than one sixteenth.

However, ^{it is} not, and it isfoundby a very simple calculation that this probability ^is greater than one fourteenth. Indeed it would be the fourth power

A

of , of

|, and_{of |} in the _first,the second, and the third of the preceding suppositions concerning the colorsof theballsintheurn. Multiplyingrespectively each power

by

^the probability of the corresponding supposition, orby f^,f^, and ,the

sum

of the products will be the probabilityof extracting successively four black balls.

One

has thusfor this probability ^^{29 ,} a fraction greater than -fa. This paradox ^is explained

by

considering ^thatthe indication of the superiority of white balls over the black ones at the first drawing does notexclude atallthesuperiority oftheblackballs over the white ones, asuperioritywhich excludes the supposition of the _equality of the colors. But this superiority, though ^but slightly probable, ought ^to render theprobabilityof drawingsuccessivelya given

number

ofblackballs greater than^{in this} supposition

if the

number

is considerable; and one has _justseen that this

commences when

_{the given}

number

is equal tofour. Let usconsider againan urnwhich contains severalwhite and blackballs. Letussuppose^atfirst thatthereisonlyonewhiteballand oneblack. Itis

then an even betthat a white ballwill be extractedin

one_drawing. Butitseems forthe_equalityof the bet thatone

who

bets on extracting the white^ball ought to have two _drawings ifthe urn contains two black and one_white,three drawings^ifitcontainsthree black and onewhite, andsoon; itissupposed^thataftereach drawing ^{theextracted ballis}placed again ⁱⁿthe urn.

We

are convinced _easily that this first idea is

erroneous. Indeed in thecase of two black and one white ball, the probabilityof extracting two black in

two _drawings is the secondpower ^off^or^; butthis

167 probabilityaddedtothat ofdrawing^awhiteballintwo drawings ^is certaintyor unity, since ^itis certainthat twoblack balls or at least onewhite ball ought^tobe drawn; the probability ^{in this last} case is then

-|, a fractiongreater than f^. There wouldstillbe a greater advantage ^{in thebet of}drawing one^{whiteball in five} draws

when

the urn containsfive black and one white ball; this bet is even advantageous ^{in four}drawings^;

itreturns then to thatof throwing ^{six infour} throws with a_singledie.

The

Chevalier de Mere,

who

caused the invention of thecalculus of probabilitiesbyencouraging^{his friend} Pascal,the great geometrician, ^tooccupy^himselfwith

it, said to him ^''that hehadfound errorinthe

num-bersby^{this ratio. If}

we

undertaketo

make

sixwith onediethere isan advantage ⁱⁿundertaking^{it infour} throws, as 671 ^to625. ^If

we

undertaketo

make

two sixeswithtwodice, there ^is a disadvantage ⁱⁿ under-takingⁱⁿ 24^throws.

At

least 24^isto 36,the

number

ofthe faces ofthe two _dice, as 4 ^isto 6, the

number

of faces of one die."

"This

was," wrote Pascal to Fermat, ^''his great scandalwhich causedhimto say boldly^thatthe propositionswerenot constantandthat arithmetic was demented. . . .

He

_{has a verygood}

mind, buthe is not a geometrician, which _is,asyou know,^agreat^fault.

''

The

Chevalier de Mere, deceived by ^{a false} analogy, thought ^{that in} the case ofthe equality of betsthe

number

ofthrowsought^toincrease

in proportion^tothe

number

ofallthechances_possible, whichis not_exact,but whichapproaches exactness^as this

number

becomes_larger.

One

hasendeavoredtoexplain the superiorityof the

A

ESSAY

births ofboys^{over thoseof} girlsbythe general^desire of fathers to have a son

who

would perpetuate the name.

Thus

_byimagining an urn^filledwithan_infinity of white andblackballsin equalnumber, and suppos-ing a great

number

of persons each of

whom

draws a ballfrom this urnand continues with the intention of stopping

when

he shall have extracted awhite _ball, onehas believed thatthis intentionought^{torender the}

number

ofwhiteballsextracted superior to that ofthe black ones. Indeed this intention gives necessarily after all the drawings a

number

ofwhite ^balls equal tothat of persons, andit ispossible that these draw-ingswould never lead ablackball. Butit iseasy^to see thatthisfirst notion is onlyan_illusion; for ifone conceivesthatinthefirstdrawing^allthe persons draw at once a ball from the _urn, it is evident that their intention canhaveno influence upon^{the color of the} balls which ought ^to appear ^{at this} drawing. ^Its unique^effectwillbe toexclude from the second draw-ing the persons

who

shall havedrawn a whiteoneat the first. Itislikewise apparent ^thatthe intention of the persons

who

^shall take _part in the

new

_drawing

willhavenoinfluenceupon ^{the colorofthe balls}which shallbe drawn, and thatitwillbe thesame atthe fol-lowing drawings. Thisintentionwillhavenoinfluence thenupon^{the color oftheballsextractedinthe}totality of drawings^{; it}will, however, causemore orfewer to participate at each drawing.

The

ratio ofthewhite ballsextracted to the blackones will differ thus very

littlefrom_unity. Itfollows thatthe

number

_{of persons} beingsupposed verylarge,^ifobservation _givesbetween the colorsextracted a ratiowhich differssensiblyfrom

unity, ^{it is} very probable ^that the

same

difference is

foundbetween_unityandtheratio ofthe whiteballs to theblack contained inthe urn.

I count again

among

^illusionsthe application which Liebnitz and Daniel Bernoulli have

made

ofthe cal-culus of probabilities to the summation ofseries. If

one reducesthe fraction whose numeratoris unityand whose denominatorisunity plus a_variable, in a series prescribed by^{the ratioto the}powers^ofthisvariable, ^it iseasy^{to see that in} supposing the ^variable equal ^to unitythe fraction becomes , and the series becomes plus one, minus _{one, plus one,} minus one, ^etc. In adding the^firsttwo terms, thesecond two, andso_on, the series is transformed into another ofwhich each term is zero. Grandi, an Italian _Jesuit, concluded from this the possibility ofthe creation; because the seriesbeing always ^, he sawthis fraction spring from an _infinityof zerosorfrom _nothing. Itwas thusthat Liebnitz believed hesawthe image ^{of creationin his} binary ^arithmetic where he employed only the two characters, unity and zero.

He

_imagined, since

God

can be represented

by

unityand _nothingby^{zero, that} the

Supreme

Being had drawn fromnothing^allbeings, as unity with zero expresses ^all the numbers in this

system of arithmetic. This idea was so pleasing ^to Liebnitz that he communicated it to the _Jesuit Grimaldi, president ^of the tribunal ofmethematicsin China, ⁱⁿ thehope^thatthis

emblem

of creation would convertto Christianitythe emperor^there

who

particu-larlyloved the sciences. I report ^this incident only to

show

towhat ^extent the prejudices of infancycan misleadthe greatestmen.

17

A

Liebnitz, always ^led by^a singular and _very loose metaphysics, considered^thatthe seriesplusone,minus one, plusone, etc.,becomes_{unity or}zero according^as one_{stops at}a

number

of termsoddoreven; and asin infinity there is no reason to preferthe even

number

tothe odd, one oughtfollowing the^rulesofprobability, totake thehalf oftheresults relative tothesetwokinds ofnumbers, and which^arezeroand _unity,which_gives

for the value of the series. Daniel Bernoulli has since extended this reasoning ^to the summation of series formedfrom_periodicterms. Butalltheseseries

havenovalues properly speaking^; they get

them

_only

in the case where their terms are multiplied by ^the successivepowers^{ofavariablelessthan} unity.

Then

theseseriesarealwaysconvergent,however smallone supposes the^differenceof the variable from_unity; and

it is easy^to demonstrate that the values assignedby Bernoulli, by^{virtueoftheruleof}probabilities, arethe

same

values of the generative ^fraction of the _series,

when

one _supposesin thesefractionsthevariableequal to unity. Thesevaluesareagain the ^limitswhichthe series approach

more

and _more, in proportion^as the variable approaches unity. But

when

the variable is

exactly equalto unitytheseriesceasetobe convergent;

theyhavevalues only^asfarasone arreststhem.

The

remarkable ratio ofthis application of the calculusof probabilities with the limits ofthevalues of _periodic series supposes^thatthe terms of theseseriesare multi-plied by ^{all the} consecutive powers ^{of the variable.}

Butthisseries

may

^{resultfrom the}development ^ofan

infinityofdifferent fractions inwhichthisdidnotoccur.

Thus

the series plus one, minus_{-one, plus} one, etc.,

i?i

may

spring from thedevelopment^ofafractionwhose numerator is unity plus the _variable, and whose denominatoristhisnumerator augmented bythe square of the variable. Supposing^thevariableequalto unity, this developmentchanges, ⁱⁿ theseries proposed, and the generative ^fraction becomes _equal tof^{; therules} of probabilitieswould _give then a false result, which proves

how

_dangerous itwould be to employ ^similar reasoning, especially ⁱⁿ the mathematical sciences, which ought^tobeespecially distinguished

by

^therigor of their operations.

We

are led naturally to believe that the order according ^to which

we

seethings renewed upon ^the earth has existed from all times and will continue always. Indeed if the present ^{state of} the universe were _exactly similar to the anterior state which has produced ^{it, it}would_givebirth in itsturn toasimilar state; the succession of these states would then be eternal. ^I havefound by^theapplication of analysis to the law ofuniversal gravity^thatthe

movement

of rota-tionandofrevolution ofthe planets andsatellites, and the_position of the orbitsandoftheirequators^are sub-jectedonlyto periodic inequalities. Incomparing^With ancient _eclipses the theory of the ^secular equation of the

moon

I have found that since Hipparchus ^the duration of the dayhas not variedby^thehundredthof a second, and thatthe

mean

temperature ^ofthe earth has not diminished the one-hundredth ofa degree.

Thus

the stability of actual order appears established at thesametimebytheoryand byobservations. But this orderis effectedby^{divers causes} which an

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