_CONCERNING ILLUSIONS IN THE ESTIMATION OF PROBABILITIES._
The mind has its illusions as the sense of sight; and in the same manner
that the sense of feeling corrects the latter, reflection and
calculation correct the former. Probability based upon a daily
experience, or exaggerated by fear and by hope, strikes us more than a
superior probability but it is only a simple result of calculus. Thus we
do not fear in return for small advantages to expose our life to dangers
much less improbable than the drawing of a quint in the lottery of
France; and yet no one would wish to procure for himself the same
advantages with the certainty of losing his life if this quint should be
drawn.
Our passions, our prejudices, and dominating opinions, by exaggerating
the probabilities which are favorable to them and by attenuating the
contrary probabilities, are the abundant sources of dangerous illusions.
Present evils and the cause which produced them effect us much more than
the remembrance of evils produced by the contrary cause; they prevent us
from appreciating with justice the inconveniences of the ones and the
others, and the probability of the proper means to guard ourselves
against them. It is this which leads alternately to despotism and to
anarchy the people who are driven from the state of repose to which they
never return except after long and cruel 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 principal cause of error against which one cannot
sufficiently guard himself.
It is principally at games of chance that a multitude of illusions
support hope and sustain it against unfavorable chances. The majority of
those who play at lotteries do not know how many chances are to their
advantage, how many are contrary to them. They see only the possibility
by a small stake of gaining a considerable sum, and the projects which
their imagination brings forth, exaggerate to their eyes the probability
of obtaining it; the poor man especially, excited by the desire of a
better fate, risks at play his necessities by clinging to the most
unfavorable combinations which promise him a great benefit. All would be
without doubt surprised by the immense number of stakes lost if they
could know of them; but one takes care on the contrary to give to the
winnings a great publicity, which becomes a new cause of excitement for
this funereal play.
When a number in the lottery of France has not been drawn for a long
time the crowd is eager to cover it with stakes. They judge since the
number has not been drawn for a long time that it ought at the next
drawing to be drawn in preference to others. So common an error appears
to me to rest upon an illusion by which one is carried back
involuntarily to the origin of events. It is, for example, very
improbable that at the play of heads and tails one will throw heads ten
times in succession. This improbability which strikes us indeed when it
has happened nine times, leads us to believe that at the tenth throw
tails will be thrown. But the past indicating in the coin a greater
propensity for heads than for tails renders the first of the events more
probable than the second; it increases as one has seen the probability
of throwing heads at the following throw. A similar illusion persuades
many people that one can certainly win in a lottery by placing each time
upon the same number, until it is drawn, a stake whose product surpasses
the sum of all the stakes. But even when similar speculations would not
often be stopped by the impossibility of sustaining them they would not
diminish the mathematical disadvantage of speculators and they would
increase their moral disadvantage, since at each drawing they would risk
a very large part of their fortune.
I have seen men, ardently desirous of having a son, who could learn only
with anxiety of the births of boys in the month when they expected to
become fathers. Imagining that the ratio of these births to those of
girls ought to be the same at the end of each month, they judged that
the boys already born would render more probable the births next of
girls. Thus the extraction of a white ball from an urn which contains a
limited number of white balls and of black balls increases the
probability of extracting a black ball at the following drawing. But
this ceases to take place when the number of balls in the urn is
unlimited, as one must suppose in order to compare this case with that
of births. If, in the course of a month, there were born many more boys
than girls, one might suspect that toward the time of their conception a
general cause had favored masculine conception, which would render more
probable the birth next of a boy. The irregular events of nature are not
exactly comparable to the drawing of the numbers of a lottery in which
all the numbers are mixed at each drawing in such a manner as to render
the chances of their drawing perfectly equal. The frequency of one of
these events seems to indicate a cause slightly favoring it, which
increases the probability of its next return, and its repetition
prolonged for a long time, such as a long series of rainy days, may
develop unknown causes for its change; so that at each expected event we
are not, as at each drawing of a lottery, led back to the same state of
indecision in regard to what ought to happen. But in proportion as the
observation of these events is multiplied, the comparison of their
results with those of lotteries becomes more exact.
By an illusion contrary to the preceding ones one seeks in the past
drawings of the lottery of France the numbers most often drawn, in order
to form combinations upon which one thinks to place the stake to
advantage. But when the manner in which the mixing of the numbers in
this lottery is considered, the past ought to have no influence upon the
future. The very frequent drawings of a number are only the anomalies of
chance; I have submitted several of them to calculation and have
constantly found that they are included within the limits which the
supposition of an equal possibility of the drawing of all the numbers
allows us to admit without improbability.
In a long series of events of the same kind the single chances of hazard
ought sometimes to offer the singular veins of good luck or bad luck
which the majority of players do not fail to attribute to a kind of
fatality. It happens often in games which depend at the same time upon
hazard and upon the competency of the players, that that one who loses,
troubled by his loss, seeks to repair it by hazardous throws which he
would shun in another situation; thus he aggravates his own ill luck and
prolongs its duration. It is then that prudence becomes necessary and
that it is of importance to convince oneself that the moral disadvantage
attached to unfavorable chances is increased by the ill luck itself.
The opinion that man has long been placed in the centre of the universe,
considering himself the special object of the cares of nature, leads
each individual to make himself the centre of a more or less extended
sphere and to believe that hazard has preference for him. Sustained by
this belief, players often risk considerable sums at games when they
know that the chances are unfavorable. In the conduct of life a similar
opinion may sometimes have advantages; but most often it leads to
disastrous enterprises. Here as everywhere illusions are dangerous and
truth alone is generally useful.
One of the great advantages of the calculus of probabilities is to teach
us to distrust first opinions. As we recognize that they often deceive
when they may be submitted to calculus, we ought to conclude that in
other matters confidence should be given only after extreme
circumspection. Let us prove this by example.
An urn contains four balls, black and white, but which are not all of
the same color. One of these balls has been drawn whose color is white
and which has been put back in the urn in order to proceed again to
similar drawings. One demands the probability of extracting only black
balls in the four following drawings.
If the white and black were in equal number this probability would be
the fourth power of the probability ½ of extracting a black ball at each
drawing; it would be then 1/16. But the extraction of a white ball at
the first drawing indicates a superiority in the number of white balls
in the urn; for if one supposes in the urn three white balls and one
black the probability of extracting a white ball is ¾; it is 2/4 if one
supposes two white balls and two black; finally it is reduced to ¼ if
one supposes three black balls and one white. Following the principle of
the probability of causes drawn from events the probabilities of these
three suppositions are among themselves as the quantities ¾, 2/4, ¼;
they are consequently equal to 3/6, 2/6, ⅙. It is thus a bet of 5
against 1 that the number of black balls is inferior, or at the most
equal, to that of the white. It seems then that after the extraction of
a white ball at the first drawing, the probability of extracting
successively four black balls ought to be less than in the case of the
equality of the colors or smaller than one sixteenth. However, it is
not, and it is found by a very simple calculation that this probability
is greater than one fourteenth. Indeed it would be the fourth power of
¼, of 2/4, and of ¾ in the first, the second, and the third of the
preceding suppositions concerning the colors of the balls in the urn.
Multiplying respectively each power by the probability of the
corresponding supposition, or by 3/6, 2/6, and ⅙, the sum of the
products will be the probability of extracting successively four black
balls. One has thus for this probability 29/384, a fraction greater than
1/14. This paradox is explained by considering that the indication of
the superiority of white balls over the black ones at the first drawing
does not exclude at all the superiority of the black balls over the
white ones, a superiority which excludes the supposition of the equality
of the colors. But this superiority, though but slightly probable, ought
to render the probability of drawing successively a given number of
black balls greater than in this supposition if the number is
considerable; and one has just seen that this commences when the given
number is equal to four. Let us consider again an urn which contains
several white and black balls. Let us suppose at first that there is
only one white ball and one black. It is then an even bet that a white
ball will be extracted in one drawing. But it seems for the equality of
the bet that one who bets on extracting the white ball ought to have two
drawings if the urn contains two black and one white, three drawings if
it contains three black and one white, and so on; it is supposed that
after each drawing the extracted ball is placed again in the urn.
We are convinced easily that this first idea is erroneous. Indeed in the
case of two black and one white ball, the probability of extracting two
black in two drawings is the second power of ⅔ or 4/9; but this
probability added to that of drawing a white ball in two drawings is
certainty or unity, since it is certain that two black balls or at least
one white ball ought to be drawn; the probability in this last case is
then 5/9, a fraction greater than ½. There would still be a greater
advantage in the bet of drawing one white ball in five draws when the
urn contains five black and one white ball; this bet is even
advantageous in four drawings; it returns then to that of throwing six
in four throws with a single die.
The Chevalier de Meré, who caused the invention of the calculus of
probabilities by encouraging his friend Pascal, the great geometrician,
to occupy himself with it, said to him "that he had found error in the
numbers by this ratio. If we undertake to make six with one die there is
an advantage in undertaking it in four throws, as 671 to 625. If we
undertake to make two sixes with two dice, there is a disadvantage in
undertaking in 24 throws. At least 24 is to 36, the number of the faces
of the two dice, as 4 is to 6, the number of faces of one die." "This
was," wrote Pascal to Fermat, "his great scandal which caused him to say
boldly that the propositions were not constant and that arithmetic was
demented.... He has a very good mind, but he is not a geometrician,
which is, as you know, a great fault." The Chevalier de Meré, deceived
by a false analogy, thought that in the case of the equality of bets the
number of throws ought to increase in proportion to the number of all
the chances possible, which is not exact, but which approaches exactness
as this number becomes larger.
One has endeavored to explain the superiority of the births of boys over
those of girls by the general desire of fathers to have a son who would
perpetuate the name. Thus by imagining an urn filled with an infinity of
white and black balls in equal number, and supposing a great number of
persons each of whom draws a ball from this urn and continues with the
intention of stopping when he shall have extracted a white ball, one has
believed that this intention ought to render the number of white balls
extracted superior to that of the black ones. Indeed this intention
gives necessarily after all the drawings a number of white balls equal
to that of persons, and it is possible that these drawings would never
lead a black ball. But it is easy to see that this first notion is only
an illusion; for if one conceives that in the first drawing all the
persons draw at once a ball from the urn, it is evident that their
intention can have no influence upon the color of the balls which ought
to appear at this drawing. Its unique effect will be to exclude from the
second drawing the persons who shall have drawn a white one at the
first. It is likewise apparent that the intention of the persons who
shall take part in the new drawing will have no influence upon the color
of the balls which shall be drawn, and that it will be the same at the
following drawings. This intention will have no influence then upon the
color of the balls extracted in the totality of drawings; it will,
however, cause more or fewer to participate at each drawing. The ratio
of the white balls extracted to the black ones will differ thus very
little from unity. It follows that the number of persons being supposed
very large, if observation gives between the colors extracted a ratio
which differs sensibly from unity, it is very probable that the same
difference is found between unity and the ratio of the white balls to
the black contained in the urn.
I count again among illusions the application which Liebnitz and Daniel
Bernoulli have made of the calculus of probabilities to the summation of
series. If one reduces the fraction whose numerator is unity and whose
denominator is unity plus a variable, in a series prescribed by the
ratio to the powers of this variable, it is easy to see that in
supposing the variable equal to unity the fraction becomes ½, and the
series becomes plus one, minus one, plus one, minus one, etc. In adding
the first two terms, the second two, and so on, the series is
transformed into another of which each term is zero. Grandi, an Italian
Jesuit, concluded from this the possibility of the creation; because the
series being always ½, he saw this fraction spring from an infinity of
zeros or from nothing. It was thus that Liebnitz believed he saw the
image of creation in his binary arithmetic where he employed only the
two characters, unity and zero. He imagined, since God can be
represented by unity and nothing by zero, that the Supreme Being had
drawn from nothing all beings, 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 of mathematics in China, in the hope that this emblem of
creation would convert to Christianity the emperor there who
particularly loved the sciences. I report this incident only to show to
what extent the prejudices of infancy can mislead the greatest men.
Liebnitz, always led by a singular and very loose metaphysics,
considered that the series plus one, minus one, plus one, etc., becomes
unity or zero according as one stops at a number of terms odd or even;
and as in infinity there is no reason to prefer the even number to the
odd, one ought following the rules of probability, to take the half of
the results relative to these two kinds of numbers, and which are zero
and unity, which gives ½ for the value of the series. Daniel Bernoulli
has since extended this reasoning to the summation of series formed from
periodic terms. But all these series have no values properly speaking;
they get them only in the case where their terms are multiplied by the
successive powers of a variable less than unity. Then these series are
always convergent, however small one supposes the difference of the
variable from unity; and it is easy to demonstrate that the values
assigned by Bernoulli, by virtue of the rule of probabilities, are the
same values of the generative fraction of the series, when one supposes
in these fractions the variable equal to unity. These values are again
the limits which the series approach more and more, in proportion as the
variable approaches unity. But when the variable is exactly equal to
unity the series cease to be convergent; they have values only as far as
one arrests them. The remarkable ratio of this application of the
calculus of probabilities with the limits of the values of periodic
series supposes that the terms of these series are multiplied by all the
consecutive powers of the variable. But this series may result from the
development of an infinity of different fractions in which this did not
occur. Thus the series plus one, minus one, plus one, etc., may spring
from the development of a fraction whose numerator is unity plus the
variable, and whose denominator is this numerator augmented by the
square of the variable. Supposing the variable equal to unity, this
development changes, in the series proposed, and the generative fraction
becomes equal to ⅔; the rules of probabilities would give then a false
result, which proves how dangerous it would be to employ similar
reasoning, especially in the mathematical sciences, which ought to be
especially distinguished by the rigor of their operations.
We are led naturally to believe that the order according to which we see
things 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
give birth in its turn to a similar state; the succession of these
states would then be eternal. I have found by the application of
analysis to the law of universal gravity that the movement of rotation
and of revolution of the planets and satellites, and the position of the
orbits and of their equators are subjected only to periodic
inequalities. In comparing with ancient eclipses the theory of the
secular equation of the moon I have found that since Hipparchus the
duration of the day has not varied by the hundredth of a second, and
that the mean temperature of the earth has not diminished the
one-hundredth of a degree. Thus the stability of actual order appears
established at the same time by theory and by observations. But this
order is effected by divers causes which an attentive examination
reveals, and which it is impossible to submit to calculus.
The actions of the ocean, of the atmosphere, and of meteors, of
earthquakes, and the eruptions of volcanoes, agitate continually the
surface of the earth and ought to effect in the long run great changes.
The temperature of climates, the volume of the atmosphere, and the
proportion of the gases which constitute it, may vary in an
inappreciable manner. The instruments and the means suitable to
determine these variations being new, observation has been unable up to
this time to teach us anything in this regard. But it is hardly probable
that the causes which absorb and renew the gases constituting the air
maintain exactly their respective proportions. A long series of
centuries will show the alterations which are experienced by all these
elements so essential to the conservation of organized beings. Although
historical monuments do not go back to a very great antiquity they offer
us nevertheless sufficiently great changes which have come about by the
slow and continued action of natural agents. Searching in the bowels of
the earth one discovers numerous débris of former nature, entirely
different from the present. Moreover, if the entire earth was in the
beginning fluid, as everything appears to indicate, one imagines that in
passing from that state to the one which it has now, its surface ought
to have experienced prodigious changes. The heavens itself in spite of
the order of its movements, is not unchangeable. The resistance of light
and of other ethereal fluids, and the attraction of the stars ought,
after a great number of centuries, to alter considerably the planetary
movements. The variations already observed in the stars and in the form
of the nebulæ give us a presentiment of those which time will develop in
the system of these great bodies. One may represent the successive
states of the universe by a curve, of which time would be the abscissa
and of which the ordinates are the divers states. Scarcely knowing an
element of this curve we are far from being able to go back to its
origin; and if in order to satisfy the imagination, always restless from
our ignorance of the cause of the phenomena which interest it, one
ventures some conjectures it is wise to present them only with extreme
reserve.
There exists in the estimation of probabilities a kind of illusions,
which depending especially upon the laws of the intellectual
organization demands, in order to secure oneself against them, a
profound examination of these laws. The desire to penetrate into the
future and the ratios of some remarkable events, to the predictions of
astrologers, of diviners and soothsayers, to presentiments and dreams,
to the numbers and the days reputed lucky or unlucky, have given birth
to a multitude of prejudices still very widespread. One does not reflect
upon the great number of non-coincidences which have made no impression
or which are unknown. However, it is necessary to be acquainted with
them in order to appreciate the probability of the causes to which the
coincidences are attributed. This knowledge would confirm without doubt
that which reason tells us in regard to these prejudices. Thus the
philosopher of antiquity to whom is shown in a temple, in order to exalt
the power of the god who is adored there, the _ex veto_ of all those who
after having invoked it were saved from shipwreck, presents an incident
consonant with the calculus of probabilities, observing that he does not
see inscribed the names of those who, in spite of this invocation, have
perished. Cicero has refuted all these prejudices with much reason and
eloquence in his _Treatise on Divination_, which he ends by a passage
which I shall cite; for one loves to find again among the ancients the
thunderbolts of reason, which, after having dissipated all the
prejudices by its light, shall become the sole foundation of human
institutions.
"It is necessary," says the Roman orator, "to reject divination by
dreams and all similar prejudices. Widespread superstition has
subjugated the majority of minds and has taken possession of the
feebleness of men. It is this we have expounded in our books upon the
nature of the gods and especially in this work, persuaded that we shall
render a service to others and to ourselves if we succeed in destroying
superstition. However (and I desire especially in this regard my thought
be well comprehended), in destroying superstition I am far from wishing
to disturb religion. Wisdom enjoins us to maintain the institutions and
the ceremonies of our ancestors, touching the cult of the gods.
Moreover, the beauty of the universe and the order of celestial things
force us to recognize some superior nature which ought to be remarked
and admired by the human race. But as far as it is proper to propagate
religion, which is joined to the knowledge of nature, so far it is
necessary to work toward the extirpation of superstition, for it
torments one, importunes one, and pursues one continually and in all
places. If one consult a diviner or a soothsayer, if one immolates a
victim, if one regards the flight of a bird, if one encounters a
Chaldean or an aruspex, if it lightens, if it thunders, if the
thunderbolt strikes, finally, if there is born or is manifested a kind
of prodigy, things one of which ought often to happen, then superstition
dominates and leaves no repose. Sleep itself, this refuge of mortals in
their troubles and their labors, becomes by it a new source of
inquietude and fear."
All these prejudices and the terrors which they inspire are connected
with physiological causes which continue sometimes to operate strongly
after reason has disabused us of them. But the repetition of acts
contrary to these prejudices can always destroy them.