CONCERNING THE PROBABILITIES OF TESTI- TESTI-MONIES

THE

majority of our opinions being foundedonthe probability of proofs^itis indeed important^tosubmit it

to calculus. Things^{itistrueoften}

become

_impossible by ^the difficulty of appreciating the veracity of wit-nessesand bythe great

number

ofcircumstanceswhich

accompany

^the deeds they ^attest; but one is able in severalcases toresolvetheproblems which have

much

analogy ^{with the} questions which are proposed and whose solutions

may

be regarded^{as suitable} approxi-mationstoguideandtodefendusagaint the^errorsand the dangers^offalsereasoning^towhich

we

areexposed.

An

approximation ^{ofthis kind,}

when

itiswellmade,

is always preferable tothe most _specious reasonings.

Let_{us try}thento give

some

_general rules for obtain-ing^it.

A

_single

number

hasbeen drawnfroman urnwhich contains a thousand ofthem.

A

witnesstothis draw-ingannounces that

number

₇₉is drawn; oneasks the probabilityofdrawing ^this number. Let us suppose that experience has

made known

that this witness

log

no A

PHILOSOPHICAL ESSAY

ON

PROBABILITIES.

deceives onetimein ten, so thatthe probability of^his testimony^isT

V

^{Here the eventobserved is the}

wit-ness_attesting that

number

₇₉ is drawn. This event

may

^{resultfrom thetwo}followinghypotheses,namely:

that thewitness utters the truth or thathe deceives.

Following ^the principle thathas been expounded on the probability ofcausesdrawn from events observed

itisnecessary^firsttodetermine a _priorithe probabil-ity of the eventin each hypothesis. ^In the first, the probability that the witness will announce

number

₇₉

isthe probability^itself of the drawing^{of this}number, that isto say, TTTOTT- Itisnecessary^to multiply^it by the probability j

6

ffof the _veracityof the witness_; one will have then T |hn5f r the probabilityof the event observedinthis hypothesis. ^Ifthewitness _deceives,

number

₇₉ is not drawn, and ^the probability of this case is $$$$. But to announce the drawing ^{of this}

number

the witness has to chooseit

among

^the 999 numbersnotdrawn; andashe issupposed^to have no motive of preference ^for the ones rather than the others, the probability that he willchoose

number

₇₉

is -577; multiplying, then, ^thisprobability by^the pre-ceding one,

we

shallhave _y^Vo f rtheprobability that the witness will announce

number

₇₉ in the second hypothesis. ^{It is} necessary again ^to multiply ^this probabilitybyT

V

^{of the} hypothesis itself, which_gives uriinr ^{f r} t^ e probabilityof the event relative to this hypothesis.

Now

if

we

form afractionwhose numera-toristheprobability relative tothefirsthypothesis,and whose denominatoristhe

sum

oftheprobabilities rela-tivetothe twohypotheses,

we

shall have,by^thesixth principle, the probability of the first hypothesis, and

CONCERNING THE PROBABILITIES

OF

TESTIMONIES,

m

thisprobability will be_T⁹_ff; thatis to say,the _veracity itselfof thewitness. Thisis likewise the probability ofthe drawing^of

number

_79.

The

probabilityof the falsehood of the witnessandofthe failure ofdrawing

this

number

^isfa.

Ifthe _witness, wishingto deceive, has

some

interest in choosing

number

₇₉

among

^{thenumbersnot}drawn,

ifhe_judges, for example, ^that having placed upon

this

number

a considerable _stake, the announcement ofits drawing^{will increase his} credit, the probability that hewill choosethis

number

will no _{longer be} as at_{first, -jfg,}itwill thenbe , , etc., according ^tothe interest that hewillhave in announcing ^its drawing.

Supposing ^{it to be}|, ^itwill be _necessary to multiply

by

^{thisfractionthe} probabilityTVVo

m

orderto getⁱⁿ the hypothesis^ofthe falsehood the probabilityof the event observed, whichitis necessary^{still to} multiply by y^, which _{gives TihhjT} f r the probability of the eventⁱⁿthe second hypothesis.

Then

the probability ofthe firsthypothesis,^orof thedrawing ^of

number

_79,

is reduced by^the preceding ^{rule to} yfg-. It is then very

much

decreased

by

the consideration of the in-terest which the witness

may

^{have in}announcing^the drawing ^of

number

_79. In truth this same interest increasesthe probability-^^thatthe witnesswillspeak thetruthif

number

₇₉is drawn. Butthis probability cannot exceed unity or | ; thus the probability ofthe drawing^of

number

₇₉will not surpassTy>T-

Common

sensetells usthatthis interestoughtto inspiredistrust, butcalculus appreciates theinfluence ofit.

The

probability a priori of the

number

announced

by

the witness is unitydivided by^the

number

of the

H2 A

PHILOSOPHICAL ESSAY

ON

PROBABILITIES.

numbers in the _urn; it is changed by ^{virtue of the} proof^intothe_veracityitselfofthe _witness;it

may

^then

be decreased

by

^the proof. ^{If, for} example, ^{the urn} contains only two numbers, which _gives for the probability apriori of the drawing ^of

number

^I, andif

the _veracity ofa witness

who

announces itis T%, ^this drawing becomes ^lessprobable. Indeed itis apparent, since thewitness has then

more

inclination towards a falsehood than towards the truth, that his testimony ought ^to decrease the probabilityof the fact attested every time^{that this} probabilityequals or surpasses ^. Butifthere arethreenumbersin the urn theprobability a priori ^ofthe drawing ^of

number

I is increased by theaffirmation of awitnesswhose_{veracitysurpasses} .

Suppose

now

thatthe urn contains 999^{black balls} and one white ball, and that one ball having been drawn a witness ofthe drawing announces ^{that this} ball iswhite.

The

probability ofthe event observed, determineda_prioriinthefirsthypothesis,^willbe_here, as in the preceding question, equal^to -foooir- But

m

the hypothesis where the witness_deceives, the white ball is not drawn and the probability of this case

is TV(TV ^{It ls} necessary ^to multiply^it by ^the prob-abilityT

V

^{of the} falsehood, which gives T|||^ ^for the probabilityof the event observed relativetothe second hypothesis. This probability was _{only T ol7nr}

m

tne preceding question; ^this great ^{difference results} from this thatablack^ballhaving been drawn ^thewitness

who

wishes to deceive has no choice atall to

make

among

^the 999^ballsnot drawnin orderto announce the drawing ^{of a white ball.}

Now

ifone forms two fractionswhosenumeratorsaretheprobabilities relative

113 toeach hypothesis,andwhose

common

denominatoris

the

sum

ofthese probabilities, onewill have

i^^

^for

theprobability of the^firsthypothesisandofthedrawing ofa white _ball, and _TVir⁹_ff f r the probability of the second hypothesis andof the drawing ^{ofablackball.}

Thislast probabilitystronglyapproaches certainty; it

would approach ^it

much

nearer and would

become

TVoVoVs ^{ifthe urn} contained a million balls of which one was_white, the drawing^{of awhite ball} becoming then

much

more extraordinary.

We

seethus

how

the probabilityof the falsehood increases in the measure thatthe deed becomes

more

extraordinary.

We

have_supposed up^{to this timethat} the witness was not mistaken at all; butifone _{admits, however,} the chance of his error the extraordinary ^incident becomes more improbable.

Then

in place of thetwo hypotheses one will have the four following ones, namely: ^thatof the witness not deceivingand_{not being} mistaken at all

; that of the witness not deceiving^at all and being mistaken^; the hypothesis of the witness deceiving and not being mistaken ^at all; finally, that ofthe witness deceivingand being mistaken. Deter-mining a priori ⁱⁿeach ofthese hypotheses the prob-ability of the event observed,

we

find

by

^{the sixth}

principlethe probability that the fact attested is false equal^toafractionwhose numerator isthe

number

of black balls in the urn _multiplied by ^the

sum

of the probabilities that thewitness does not deceive ^at all

and ismistaken, or that he deceives and is not mis-taken, and whose denominator is this numerator augmented by ^the

sum

of the probabilities that the witnessdoes not deceiveatall and isnot mistaken at

H4 A ON

PROBABILITIES.

all, or that he deceives and is mistaken atthe same time.

We

see by ^{this that if the}

number

of black ballsinthe urnisvery great, whichrenders the draw-ing^ofthewhiteballextraordinary, the probability that thefactattested isnottrue approaches most_nearly to certainty.

Applying^{this conclusiontoall} extraordinary deeds

itresults fromitthatthe probability oftheerror orof the falsehood of the witness becomesas

much

_greater

as the fact attested is

more

extraordinary.

Some

authorshave advanced the contraryonthisbasis that theviewofanextraordinary^factbeingperfectly similar tothat of an ordinary^factthe

same

motives ought ^to leadus to give the witness the

same

credence

when

he affirms the one or the other of these facts. Simple

common

^senserejectssuch a strange^assertion;but the calculus of probabilities, while confirming the findings of

common

sense,appreciates thegreatestimprobability oftestimonies inregard^toextraordinary^facts.

These authors insist and _suppose two witnesses equallyworthy^ofbelief, of

whom

the first atteststhat hesaw an individual deadfifteen daysago

whom

the second witness affirms to have seen yesterday ^full of life.

The

oneor the otherof these facts offersno improbability.

The

reservation of theindividual isa resultoftheircombination; but the testimoniesdonot bring ^{us at all} directly to this result, although the credencewhich is due these testimonies ought^notto be decreasedby^{the factthatthe result oftheir}

com-binationisextraordinary.

But if the conclusionwhich results from the

com-bination of the testimonieswas _impossible oneofthem

CONCERNING THE 115 wouldbe necessarilyfalse; but an impossible conclu-sion is the limitofextraordinaryconclusions, as error is thelimitofimprobable conclusions;^the valueofthe testimonies which becomes zero in the case of an impossible conclusion ought ^{then to be} very

much

decreased in that of an extraordinary conclusion.

This is indeed confirmed by ^{the calculus of} prob-abilities.

Inorderto

make

itplain ^letus considertwo_urns,

A

and_{B, of}whichthefirst contains amillionwhiteballs andthe second amillionblackballs.

One

drawsfrom one of these urns aball, which he_puts back intothe other_urn, from which one then draws a ball.

Two

witnesses,the one ofthefirstdrawing, the other^ofthe second,^attestthattheballwhich_theyhaveseendrawn

iswhite without _indicating the urn fromwhich it has been drawn.

Each

_testimony taken alone is not improbable; and itis easy^{to see that}the probability ofthefactattested is the_veracityitselfofthewitness.

But itfollowsfromthe combination of the testimonies thata whiteballhas beenextracted fromtheurn

A

at the firstdraw, and that then placed ⁱⁿ the urn

B

it

has reappeared ^at the second draw, which is very extraordinary; ^forthissecond_urn,containing then one white ball

among

^{amillion black}balls, theprobability of drawing ^{the white ball is} yc-UFor- ^{^}n order to determine the diminution which results in the prob-ability of the thing announced

by

^{the two witnesses}

we

shall notice that the event observed is here the affirmation by^{each of}themthatthe ball which hehas seen extracted is white. Let us represent byT⁹T ^the probability that he announces the truth, which can

occur in the present case

when

thewitness does not deceive and' is not mistaken at all, and

when

he deceivesandismistaken atthe same time.

One may

form thefour followinghypotheses^:

1st.

The

first and second witness speak the ^truth.

Then

awhiteball hasatfirstbeendrawn from the urn A, and^Iheprobabilityofthis eventis

|, sincethe ball drawn al the firstdraw

may

have been drawn either from the one orthe other urn. Consequently^{the ball} drawn, placed ⁱⁿ the urn B, has reappeared ^at the second draw; the probability of ^thisevent^is

the probability ofthefact announced isthen

Multiplying ^it

by

the product of the probabilities -fa

and _y⁹_^ that the witnesses speak ^{the truth} one will

have ^nnrVoinr ^{fr the} probability of the event ob-served in thisfirsthypothesis.

2d.

The

firstwitness speaks the^truthandthesecond does _not, whetherhe deceivesandis not mistaken or he does not deceive andis mistaken.

Then

awhite ball has been drawn from the urn

A

atthe firstdraw, andtheprobability of ^this eventis .

Then

thisball having beenplacedⁱⁿthe urn

B

a blackballhasbeen drawn from it: the probability of such drawing ^is

|_o_o^0.0.0

. one has then

#{$!$

^{for the} probability of the

compound

^event. Multiplying ^it bythe product ofthetwo probabilitiesT⁹ and T

V

^{thatthe firstwitness}

speaks the ^truth and that the second does _not, one will have

y^^fjfo.

^{for the} probability^forthe event observedinthe secondhypothesis.

3d.

The

firstwitness does not speak the ^truth and thesecond announces^it.

Then

a black^ball hasbeen

drawn

from the urn

B

atthefirstdrawing, and after

n?

having been _placedin theurn

A

a whiteballhasbeen drawn from this urn.

The

probabilityof the first of theseevents is andthatof thesecondisT #$S$T^{; the} probability of the

compound

^{event is then} i^^-^f.

Multiplying ^it

by

^the product of the probabilities 1³jr

and_yVthatthe firstwitness does not speak the ^truth and that the second announces it, one will have siiHfTmHta ^for tne probability of the event observed relativeto thishypothesis.

4th. Finally, neitherof thewitnesses speaks the^truth.

Then

a black ball hasbeen

drawn

from the urn

B

at thefirstdraw; then having been placed ⁱⁿ the urn

A

it has reappeared ^at the second drawing: the prob-abilityofthis

compound

^eventis aooooo?- Multiply-ing^itbythe product of theprobabilities-faand_y¹

^-that

each witness does not speak the ^truthonewill have 200000200 ^{f r}tneprobabilityofthe event observedin thishypothesis.

Now

^inorderto obtain the probabilityofthe thing announced by^the two_{witnesses, namely,} thatawhite ballhas been drawnat each draw, itis necessary^to divide the probability corresponding ^to the first hy-pothesis by ^the

sum

of the probabilities relative to the four hypotheses^; andthenone has forthis prob-ability y-g-ooooas' an extremely^{small fraction.}

If the two witnesses affirm the _first, that a white ballhas been drawnfrom one ofthetwo urns

A

and B; ^the second that a white ball has been likewise drawn from one of the two urns A' and B', quite similar tothe first ones, the probabilityof the thing announced by^the twowitnesseswillbe the product ^of theprobabilities of their testimonies,or y\V; ^itwillthen

n8 _A

PHILOSOPHICAL ESSAY

ON

PROBABILITIES.

be at least a hundred and _eighty thousand times greater than^thepreceding one.

One

seesby^this

how

much, ^inthefirst case,the reappearance^atthesecond draw of the white ball drawn at the first draw, the extraordinary conclusion of the two testimonies de-creases thevalue ofit.

We

^would give no credence to the testimony of a

man who

shouldattesttous thatinthrowing ahundred dice into theair theyhad allfallen onthe same ^face.

If

we

had ourselves been _spectators ofthis event

we

shouldbelieveour

own

_{eyes only}afterhaving carefully examined all the circumstances, and after having broughtⁱⁿthe testimoniesofother eyesⁱⁿ ordertobe quite sure thattherehad been neither hallucinationnor deception. But after this examination

we

should not hesitate toadmititinspiteofitsextremeimprobability;

andno one would be tempted,ⁱⁿ ordertoexplain^it,to recur to adenial ofthe laws ofvision.

We

_ought to concludefromitthatthe probabilityof the constancy of the laws ofnature is forus greater than this, that the event in question has not taken place ^{at all} a probability greater than ^{that of} the majority of his-torical factswhich

we

_regard as incontestable.

One

may

judge by^thisthe

immense

_weight of testimonies necessary ^to admit a suspension ofnatural laws, and

how

_improper it would be to apply ^{to this} case the ordinary ^rules of criticism. All those

who

without offering ^this immensity ^of testimonies support ^this

when

_makingrecitals ofevents contrary^tothose _laws, decrease rather than

augment

^{the beliefwhich} they wish to inspire^{; for} then those recitals render very probable the ^{error or} the falsehood of their authors.

CONCERNING THE 119 Butthatwhich diminishesthe beliefofeducated

men

increases often thatofthe uneducated, alwaysgreedy

forthe wonderful.

Therearethings so extraordinary^that nothing can balancetheir improbability. Butthis, by^{the effectof} adominant _opinion, can be weakenedtothe point of appearing^{inferior to}theprobabilityofthe testimonies_; and

when

this opinion changes ^an absurd statement admitted unanimouslyⁱⁿ the centurywhich _{has given}

it birth offers to the following ^centuries only a

new

proof ^of the extreme influenceofthe general opinion upon ^themore enlightened minds.

Two

_great

men

of the century of Louis

XIV.

Racine and Pascal are striking examples ^{of this. It is} painful to see with what complaisance Racine, ^this admirable _painter of the

human

heart and the most _{perfect poet} thathas everlived, reports as miraculous the recovery^{of Mile.}

Perrier, a niece of Pascal and a day pupil at the monastery ^of Port-Royal; ^{it is} painful to read the reasonsby which^{Pascalseeksto}prove^thatthismiracle should be necessary^to religion ⁱⁿorderto _justifythe doctrine ofthe

monks

ofthisabbey,^atthattime perse-cuted

by

^{the Jesuits.}

The young

^{Perrier had been}

afflicted forthree yearsandahalfbya lachrymalfistula;

she touched her afflicted eye with a ^relic whichwas pretended^to beone ofthe thorns of the crownof the Saviour andshehad faith in instant recovery.

Some

days afterward the physiciansandthe surgeons^attest the recovery, and they declare ^that nature and the remedies have had no _partin it. This event, which took _placein 1656,

made

_{a great} sensation, and "all Paris rushed," says Racine, "to Port-Royal.

The

No documento A PHILOSOPHICAL ESSAY (páginas 121-138)