Of The Calculation Of Chances.
§ 1. “Probability,” says Laplace,(176) “has reference partly to our
ignorance, partly to our knowledge. We know that among three or more
events, one, and only one, must happen; but there is nothing leading us
to believe that any one of them will happen rather than the others. In
this state of indecision, it is impossible for us to pronounce with
certainty on their occurrence. It is, however, probable that any one of
these events, selected at pleasure, will not take place; because we
perceive several cases, all equally possible, which exclude its
occurrence, and only one which favors it.
“The theory of chances consists in reducing all events of the same kind
to a certain number of cases equally possible, that is, such that we are
_equally undecided_ as to their existence; and in determining the number
of these cases which are favorable to the event of which the probability
is sought. The ratio of that number to the number of all the possible
cases is the measure of the probability; which is thus a fraction,
having for its numerator the number of cases favorable to the event, and
for its denominator the number of all the cases which are possible.”
To a calculation of chances, then, according to Laplace, two things are
necessary; we must know that of several events some one will certainly
happen, and no more than one; and we must not know, nor have any reason
to expect, that it will be one of these events rather than another. It
has been contended that these are not the only requisites, and that
Laplace has overlooked, in the general theoretical statement, a
necessary part of the foundation of the doctrine of chances. To be able
(it has been said) to pronounce two events equally probable, it is not
enough that we should know that one or the other must happen, and should
have no grounds for conjecturing which. Experience must have shown that
the two events are of equally frequent occurrence. Why, in tossing up a
half-penny, do we reckon it equally probable that we shall throw cross
or pile? Because we know that in any great number of throws, cross and
pile are thrown about equally often; and that the more throws we make,
the more nearly the equality is perfect. We may know this if we please
by actual experiment, or by the daily experience which life affords of
events of the same general character, or, deductively, from the effect
of mechanical laws on a symmetrical body acted upon by forces varying
indefinitely in quantity and direction. We may know it, in short, either
by specific experience, or on the evidence of our general knowledge of
nature. But, in one way or the other, we must know it, to justify us in
calling the two events equally probable; and if we knew it not, we
should proceed as much at hap-hazard in staking equal sums on the
result, as in laying odds.
This view of the subject was taken in the first edition of the present
work; but I have since become convinced that the theory of chances, as
conceived by Laplace and by mathematicians generally, has not the
fundamental fallacy which I had ascribed to it.
We must remember that the probability of an event is not a quality of
the event itself, but a mere name for the degree of ground which we, or
some one else, have for expecting it. The probability of an event to one
person is a different thing from the probability of the same event to
another, or to the same person after he has acquired additional
evidence. The probability to me, that an individual of whom I know
nothing but his name will die within the year, is totally altered by my
being told the next minute that he is in the last stage of a
consumption. Yet this makes no difference in the event itself, nor in
any of the causes on which it depends. Every event is in itself certain,
not probable; if we knew all, we should either know positively that it
will happen, or positively that it will not. But its probability to us
means the degree of expectation of its occurrence, which we are
warranted in entertaining by our present evidence.
Bearing this in mind, I think it must be admitted, that even when we
have no knowledge whatever to guide our expectations, except the
knowledge that what happens must be some one of a certain number of
possibilities, we may still reasonably judge, that one supposition is
more probable _to us_ than another supposition; and if we have any
interest at stake, we shall best provide for it by acting conformably to
that judgment.
§ 2. Suppose that we are required to take a ball from a box, of which we
only know that it contains balls both black and white, and none of any
other color. We know that the ball we select will be either a black or a
white ball; but we have no ground for expecting black rather than white,
or white rather than black. In that case, if we are obliged to make a
choice, and to stake something on one or the other supposition, it will,
as a question of prudence, be perfectly indifferent which; and we shall
act precisely as we should have acted if we had known beforehand that
the box contained an equal number of black and white balls. But though
our conduct would be the same, it would not be founded on any surmise
that the balls were in fact thus equally divided; for we might, on the
contrary, know by authentic information that the box contained
ninety-nine balls of one color, and only one of the other; still, if we
are not told which color has only one, and which has ninety-nine, the
drawing of a white and of a black ball will be equally probable to us.
We shall have no reason for staking any thing on the one event rather
than on the other; the option between the two will be a matter of
indifference; in other words, it will be an even chance.
But let it now be supposed that instead of two there are three
colors—white, black, and red; and that we are entirely ignorant of the
proportion in which they are mingled. We should then have no reason for
expecting one more than another, and if obliged to bet, should venture
our stake on red, white, or black with equal indifference. But should we
be indifferent whether we betted for or against some one color, as, for
instance, white? Surely not. From the very fact that black and red are
each of them separately equally probable to us with white, the two
together must be twice as probable. We should in this case expect not
white rather than white, and so much rather that we would lay two to one
upon it. It is true, there might, for aught we knew, be more white balls
than black and red together; and if so, our bet would, if we knew more,
be seen to be a disadvantageous one. But so also, for aught we knew,
might there be more red balls than black and white, or more black balls
than white and red, and in such case the effect of additional knowledge
would be to prove to us that our bet was more advantageous than we had
supposed it to be. There is in the existing state of our knowledge a
rational probability of two to one against white; a probability fit to
be made a basis of conduct. No reasonable person would lay an even wager
in favor of white against black and red; though against black alone or
red alone he might do so without imprudence.
The common theory, therefore, of the calculation of chances, appears to
be tenable. Even when we know nothing except the number of the possible
and mutually excluding contingencies, and are entirely ignorant of their
comparative frequency, we may have grounds, and grounds numerically
appreciable, for acting on one supposition rather than on another; and
this is the meaning of Probability.
§ 3. The principle, however, on which the reasoning proceeds, is
sufficiently evident. It is the obvious one that when the cases which
exist are shared among several kinds, it is impossible that _each_ of
those kinds should be a majority of the whole: on the contrary, there
must be a majority against each kind, except one at most; and if any
kind has more than its share in proportion to the total number, the
others collectively must have less. Granting this axiom, and assuming
that we have no ground for selecting any one kind as more likely than
the rest to surpass the average proportion, it follows that we can not
rationally presume this of any, which we should do if we were to bet in
favor of it, receiving less odds than in the ratio of the number of the
other kinds. Even, therefore, in this extreme case of the calculation of
probabilities, which does not rest on special experience at all, the
logical ground of the process is our knowledge—such knowledge as we then
have—of the laws governing the frequency of occurrence of the different
cases; but in this case the knowledge is limited to that which, being
universal and axiomatic, does not require reference to specific
experience, or to any considerations arising out of the special nature
of the problem under discussion.
Except, however, in such cases as games of chance, where the very
purpose in view requires ignorance instead of knowledge, I can conceive
no case in which we ought to be satisfied with such an estimate of
chances as this—an estimate founded on the absolute minimum of knowledge
respecting the subject. It is plain that, in the case of the colored
balls, a very slight ground of surmise that the white balls were really
more numerous than either of the other colors, would suffice to vitiate
the whole of the calculations made in our previous state of
indifference. It would place us in that position of more advanced
knowledge, in which the probabilities, to us, would be different from
what they were before; and in estimating these new probabilities we
should have to proceed on a totally different set of data, furnished no
longer by mere counting of possible suppositions, but by specific
knowledge of facts. Such data it should always be our endeavor to
obtain; and in all inquiries, unless on subjects equally beyond the
range of our means of knowledge and our practical uses, they may be
obtained, if not good, at least better than none at all.(177)
It is obvious, too, that even when the probabilities are derived from
observation and experiment, a very slight improvement in the data, by
better observations, or by taking into fuller consideration the special
circumstances of the case, is of more use than the most elaborate
application of the calculus to probabilities founded on the data in
their previous state of inferiority. The neglect of this obvious
reflection has given rise to misapplications of the calculus of
probabilities which have made it the real opprobrium of mathematics. It
is sufficient to refer to the applications made of it to the credibility
of witnesses, and to the correctness of the verdicts of juries. In
regard to the first, common sense would dictate that it is impossible to
strike a general average of the veracity and other qualifications for
true testimony of mankind, or of any class of them; and even if it were
possible, the employment of it for such a purpose implies a
misapprehension of the use of averages, which serve, indeed, to protect
those whose interest is at stake, against mistaking the general result
of large masses of instances, but are of extremely small value as
grounds of expectation in any one individual instance, unless the case
be one of those in which the great majority of individual instances do
not differ much from the average. In the case of a witness, persons of
common sense would draw their conclusions from the degree of consistency
of his statements, his conduct under cross-examination, and the relation
of the case itself to his interests, his partialities, and his mental
capacity, instead of applying so rude a standard (even if it were
capable of being verified) as the ratio between the number of true and
the number of erroneous statements which he may be supposed to make in
the course of his life.
Again, on the subject of juries or other tribunals, some mathematicians
have set out from the proposition that the judgment of any one judge or
juryman is, at least in some small degree, more likely to be right than
wrong, and have concluded that the chance of a number of persons
concurring in a wrong verdict is diminished the more the number is
increased; so that if the judges are only made sufficiently numerous,
the correctness of the judgment may be reduced almost to certainty. I
say nothing of the disregard shown to the effect produced on the moral
position of the judges by multiplying their numbers, the virtual
destruction of their individual responsibility, and weakening of the
application of their minds to the subject. I remark only the fallacy of
reasoning from a wide average to cases necessarily differing greatly
from any average. It may be true that, taking all causes one with
another, the opinion of any one of the judges would be oftener right
than wrong; but the argument forgets that in all but the more simple
cases, in all cases in which it is really of much consequence what the
tribunal is, the proposition might probably be reversed; besides which,
the cause of error, whether arising from the intricacy of the case or
from some common prejudice or mental infirmity, if it acted upon one
judge, would be extremely likely to affect all the others in the same
manner, or at least a majority, and thus render a wrong instead of a
right decision more probable the more the number was increased.
These are but samples of the errors frequently committed by men who,
having made themselves familiar with the difficult formulæ which algebra
affords for the estimation of chances under suppositions of a complex
character, like better to employ those formulæ in computing what are the
probabilities to a person half informed about a case than to look out
for means of being better informed. Before applying the doctrine of
chances to any scientific purpose, the foundation must be laid for an
evaluation of the chances, by possessing ourselves of the utmost
attainable amount of positive knowledge. The knowledge required is that
of the comparative frequency with which the different events in fact
occur. For the purposes, therefore, of the present work, it is allowable
to suppose that conclusions respecting the probability of a fact of a
particular kind rest on our knowledge of the proportion between the
cases in which facts of that kind occur, and those in which they do not
occur; this knowledge being either derived from specific experiment, or
deduced from our knowledge of the causes in operation which tend to
produce, compared with those which tend to prevent, the fact in
question.
Such calculation of chances is grounded on an induction; and to render
the calculation legitimate, the induction must be a valid one. It is not
less an induction, though it does not prove that the event occurs in all
cases of a given description, but only that out of a given number of
such cases it occurs in about so many. The fraction which mathematicians
use to designate the probability of an event is the ratio of these two
numbers; the ascertained proportion between the number of cases in which
the event occurs and the sum of all the cases, those in which it occurs
and in which it does not occur, taken together. In playing at cross and
pile, the description of cases concerned are throws, and the probability
of cross is one-half, because if we throw often enough cross is thrown
about once in every two throws. In the cast of a die, the probability of
ace is one-sixth; not simply because there are six possible throws, of
which ace is one, and because we do not know any reason why one should
turn up rather than another—though I have admitted the validity of this
ground in default of a better—but because we do actually know, either by
reasoning or by experience, that in a hundred or a million of throws ace
is thrown in about one-sixth of that number, or once in six times.
§ 4. I say, “either by reasoning or by experience,” meaning specific
experience. But in estimating probabilities, it is not a matter of
indifference from which of these two sources we derive our assurance.
The probability of events, as calculated from their mere frequency in
past experience, affords a less secure basis for practical guidance than
their probability as deduced from an equally accurate knowledge of the
frequency of occurrence of their causes.
The generalization that an event occurs in ten out of every hundred
cases of a given description, is as real an induction as if the
generalization were that it occurs in all cases. But when we arrive at
the conclusion by merely counting instances in actual experience, and
comparing the number of cases in which A has been present with the
number in which it has been absent, the evidence is only that of the
Method of Agreement, and the conclusion amounts only to an empirical
law. We can make a step beyond this when we can ascend to the causes on
which the occurrence of A or its non-occurrence will depend, and form an
estimate of the comparative frequency of the causes favorable and of
those unfavorable to the occurrence. These are data of a higher order,
by which the empirical law derived from a mere numerical comparison of
affirmative and negative instances will be either corrected or
confirmed, and in either case we shall obtain a more correct measure of
probability than is given by that numerical comparison. It has been well
remarked that in the kind of examples by which the doctrine of chances
is usually illustrated, that of balls in a box, the estimate of
probabilities is supported by reasons of causation, stronger than
specific experience. “What is the reason that in a box where there are
nine black balls and one white, we expect to draw a black ball nine
times as much (in other words, nine times as often, frequency being the
gauge of intensity in expectation) as a white? Obviously because the
local conditions are nine times as favorable; because the hand may
alight in nine places and get a black ball, while it can only alight in
one place and find a white ball; just for the same reason that we do not
expect to succeed in finding a friend in a crowd, the conditions in
order that we and he should come together being many and difficult. This
of course would not hold to the same extent were the white balls of
smaller size than the black, neither would the probability remain the
same; the larger ball would be much more likely to meet the hand.”(178)
It is, in fact, evident that when once causation is admitted as a
universal law, our expectation of events can only be rationally grounded
on that law. To a person who recognizes that every event depends on
causes, a thing’s having happened once is a reason for expecting it to
happen again, only because proving that there exists, or is liable to
exist, a cause adequate to produce it.(179) The frequency of the
particular event, apart from all surmise respecting its cause, can give
rise to no other induction than that _per enumerationem simplicem_; and
the precarious inferences derived from this are superseded, and
disappear from the field as soon as the principle of causation makes its
appearance there.
Notwithstanding, however, the abstract superiority of an estimate of
probability grounded on causes, it is a fact that in almost all cases in
which chances admit of estimation sufficiently precise to render their
numerical appreciation of any practical value, the numerical data are
not drawn from knowledge of the causes, but from experience of the
events themselves. The probabilities of life at different ages or in
different climates; the probabilities of recovery from a particular
disease; the chances of the birth of male or female offspring; the
chances of the destruction of houses or other property by fire; the
chances of the loss of a ship in a particular voyage, are deduced from
bills of mortality, returns from hospitals, registers of births, of
shipwrecks, etc., that is, from the observed frequency not of the
causes, but of the effects. The reason is, that in all these classes of
facts the causes are either not amenable to direct observation at all,
or not with the requisite precision, and we have no means of judging of
their frequency except from the empirical law afforded by the frequency
of the effects. The inference does not the less depend on causation
alone. We reason from an effect to a similar effect by passing through
the cause. If the actuary of an insurance office infers from his tables
that among a hundred persons now living of a particular age, five on the
average will attain the age of seventy, his inference is legitimate, not
for the simple reason that this is the proportion who have lived till
seventy in times past, but because the fact of their having so lived
shows that this is the proportion existing, at that place and time,
between the causes which prolong life to the age of seventy and those
tending to bring it to an earlier close.(180)
§ 5. From the preceding principles it is easy to deduce the
demonstration of that theorem of the doctrine of probabilities which is
the foundation of its application to inquiries for ascertaining the
occurrence of a given event, or the reality of an individual fact. The
signs or evidences by which a fact is usually proved are some of its
consequences; and the inquiry hinges upon determining what cause is most
likely to have produced a given effect. The theorem applicable to such
investigations is the Sixth Principle in Laplace’s “_Essai Philosophique
sur les Probabilités_,” which is described by him as the “fundamental
principle of that branch of the Analysis of Chances which consists in
ascending from events to their causes.”(181)
Given an effect to be accounted for, and there being several causes
which might have produced it, but of the presence of which in the
particular case nothing is known; the probability that the effect was
produced by any one of these causes _is as the antecedent probability of
the cause, multiplied by the probability that the cause, if it existed,
would have produced the given effect_.
Let M be the effect, and A, B, two causes, by either of which it might
have been produced. To find the probability that it was produced by the
one and not by the other, ascertain which of the two is most likely to
have existed, and which of them, if it did exist, was most likely to
produce the effect M: the probability sought is a compound of these two
probabilities.
CASE I. Let the causes be both alike in the second respect: either A or
B, when it exists, being supposed equally likely (or equally certain) to
produce M; but let A be in itself twice as likely as B to exist, that
is, twice as frequent a phenomenon. Then it is twice as likely to have
existed in this case, and to have been the cause which produced M.
For, since A exists in nature twice as often as B, in any 300 cases in
which one or other existed, A has existed 200 times and B 100. But
either A or B must have existed wherever M is produced; therefore, in
300 times that M is produced, A was the producing cause 200 times, B
only 100, that is, in the ratio of 2 to 1. Thus, then, if the causes are
alike in their capacity of producing the effect, the probability as to
which actually produced it is in the ratio of their antecedent
probabilities.
CASE II. Reversing the last hypothesis, let us suppose that the causes
are equally frequent, equally likely to have existed, but not equally
likely, if they did exist, to produce M; that in three times in which A
occurs, it produces that effect twice, while B, in three times, produces
it only once. Since the two causes are equally frequent in their
occurrence; in every six times that either one or the other exists, A
exists three times and B three times. A, of its three times, produces M
in two; B, of its three times, produces M in one. Thus, in the whole six
times, M is only produced thrice; but of that thrice it is produced
twice by A, once only by B. Consequently, when the antecedent
probabilities of the causes are equal, the chances that the effect was
produced by them are in the ratio of the probabilities that if they did
exist they would produce the effect.
CASE III. The third case, that in which the causes are unlike in both
respects, is solved by what has preceded. For, when a quantity depends
on two other quantities, in such a manner that while either of them
remains constant it is proportional to the other, it must necessarily be
proportional to the product of the two quantities, the product being the
only function of the two which obeys that law of variation. Therefore,
the probability that M was produced by either cause, is as the
antecedent probability of the cause, multiplied by the probability that
if it existed it would produce M. Which was to be demonstrated.
Or we may prove the third case as we proved the first and second. Let A
be twice as frequent as B, and let them also be unequally likely, when
they exist, to produce M; let A produce it twice in four times, B thrice
in four times. The antecedent probability of A is to that of B as 2 to
1; the probabilities of their producing M are as 2 to 3; the product of
these ratios is the ratio of 4 to 3; and this will be the ratio of the
probabilities that A or B was the producing cause in the given instance.
For, since A is twice as frequent as B, out of twelve cases in which one
or other exists, A exists in 8 and B in 4. But of its eight cases, A, by
the supposition, produces M in only 4, while B of its four cases
produces M in 3. M, therefore, is only produced at all in seven of the
twelve cases; but in four of these it is produced by A, in three by B;
hence the probabilities of its being produced by A and by B are as 4 to
3, and are expressed by the fractions ⁴⁄₇ and ³⁄₇. Which was to be
demonstrated.
§ 6. It remains to examine the bearing of the doctrine of chances on the
peculiar problem which occupied us in the preceding chapter, namely, how
to distinguish coincidences which are casual from those which are the
result of law; from those in which the facts which accompany or follow
one another are somehow connected through causation.
The doctrine of chances affords means by which, if we knew the _average_
number of coincidences to be looked for between two phenomena connected
only casually, we could determine how often any given deviation from
that average will occur by chance. If the probability of any casual
coincidence, considered in itself, be 1/_m_, the probability that the
same coincidence will be repeated _n_ times in succession is 1/_m_n. For
example, in one throw of a die the probability of ace being ⅙; the
probability of throwing ace twice in succession will be 1 divided by the
square of 6, or ¹⁄₃₆. For ace is thrown at the first throw once in six,
or six in thirty-six times, and of those six, the die being cast again,
ace will be thrown but once; being altogether once in thirty-six times.
The chance of the same cast three times successively is, by a similar
reasoning, ¹⁄₆₃ or ¹⁄₂₁₆; that is, the event will happen, on a large
average, only once in two hundred and sixteen throws.
We have thus a rule by which to estimate the probability that any given
series of coincidences arises from chance, provided we can measure
correctly the probability of a single coincidence. If we can obtain an
equally precise expression for the probability that the same series of
coincidences arises from causation, we should only have to compare the
numbers. This, however, can rarely be done. Let us see what degree of
approximation can practically be made to the necessary precision.
The question falls within Laplace’s sixth principle, just demonstrated.
The given fact, that is to say, the series of coincidences, may have
originated either in a casual conjunction of causes or in a law of
nature. The probabilities, therefore, that the fact originated in these
two modes, are as their antecedent probabilities, multiplied by the
probabilities that if they existed they would produce the effect. But
the particular combination of chances, if it occurred, or the law of
nature if real, would certainly produce the series of coincidences. The
probabilities, therefore, that the coincidences are produced by the two
causes in question are as the antecedent probabilities of the causes.
One of these, the antecedent probability of the combination of mere
chances which would produce the given result, is an appreciable
quantity. The antecedent probability of the other supposition may be
susceptible of a more or less exact estimation, according to the nature
of the case.
In some cases, the coincidence, supposing it to be the result of
causation at all, must be the result of a known cause; as the succession
of aces, if not accidental, must arise from the loading of the die. In
such cases we may be able to form a conjecture as to the antecedent
probability of such a circumstance from the characters of the parties
concerned, or other such evidence; but it would be impossible to
estimate that probability with any thing like numerical precision. The
counter-probability, however, that of the accidental origin of the
coincidence, dwindling so rapidly as it does at each new trial, the
stage is soon reached at which the chance of unfairness in the die,
however small in itself, must be greater than that of a casual
coincidence; and on this ground, a practical decision can generally be
come to without much hesitation, if there be the power of repeating the
experiment.
When, however, the coincidence is one which can not be accounted for by
any known cause, and the connection between the two phenomena, if
produced by causation, must be the result of some law of nature hitherto
unknown; which is the case we had in view in the last chapter; then,
though the probability of a casual coincidence may be capable of
appreciation, that of the counter-supposition, the existence of an
undiscovered law of nature, is clearly unsusceptible of even an
approximate valuation. In order to have the data which such a case would
require, it would be necessary to know what proportion of all the
individual sequences or co-existences occurring in nature are the result
of law, and what proportion are mere casual coincidences. It being
evident that we can not form any plausible conjecture as to this
proportion, much less appreciate it numerically, we can not attempt any
precise estimation of the comparative probabilities. But of this we are
sure, that the detection of an unknown law of nature—of some previously
unrecognized constancy of conjunction among phenomena—is no uncommon
event. If, therefore, the number of instances in which a coincidence is
observed, over and above that which would arise on the average from the
mere concurrence of chances, be such that so great an amount of
coincidences from accident alone would be an extremely uncommon event;
we have reason to conclude that the coincidence is the effect of
causation, and may be received (subject to correction from further
experience) as an empirical law. Further than this, in point of
precision, we can not go; nor, in most cases, is greater precision
required, for the solution of any practical doubt.(182)