PROBABILITY
Sec. 1. Chance was once believed to be a distinct power in the world,
disturbing the regularity of Nature; though, according to Aristotle, it
was only operative in occurrences below the sphere of the moon. As,
however, it is now admitted that every event in the world is due to some
cause, if we can only trace the connection, whilst nevertheless the
notion of Chance is still useful when rightly conceived, we have to find
some other ground for it than that of a spontaneous capricious force
inherent in things. For such a conception can have no place in any
logical interpretation of Nature: it can never be inferred from a
principle, seeing that every principle expresses an uniformity; nor,
again, if the existence of a capricious power be granted, can any
inference be drawn from it. Impossible alike as premise and as
conclusion, for Reason it is nothing at all.
Every event is a result of causes: but the multitude of forces and the
variety of collocations being immeasurably great, the overwhelming
majority of events occurring about the same time are only related by
Causation so remotely that the connection cannot be followed. Whilst my
pen moves along the paper, a cab rattles down the street, bells in the
neighbouring steeple chime the quarter, a girl in the next house is
practising her scales, and throughout the world innumerable events are
happening which may never happen together again; so that should one of
them recur, we have no reason to expect any of the others. This is
Chance, or chance coincidence. The word Coincidence is vulgarly used
only for the inexplicable concurrence of _interesting_ events--"quite a
coincidence!"
On the other hand, many things are now happening together or coinciding,
that will do so, for assignable reasons, again and again; thousands of
men are leaving the City, who leave at the same hour five days a week.
But this is not chance; it is causal coincidence due to the custom of
business in this country, as determined by our latitude and longitude
and other circumstances. No doubt the above chance
coincidences--writing, cab-rattling, chimes, scales, etc.--are
causally connected at some point of past time. They were predetermined
by the condition of the world ten minutes ago; and that was due to
earlier conditions, one behind the other, even to the formation of the
planet. But whatever connection there may have been, we have no such
knowledge of it as to be able to deduce the coincidence, or calculate
its recurrence. Hence Chance is defined by Mill to be: Coincidence
giving no ground to infer uniformity.
Still, some chance coincidences do recur according to laws of their own:
I say _some_, but it may be all. If the world is finite, the possible
combinations of its elements are exhaustible; and, in time, whatever
conditions of the world have concurred will concur again, and in the
same relation to former conditions. This writing, that cab, those
chimes, those scales will coincide again; the Argonautic expedition, and
the Trojan war, and all our other troubles will be renewed. But let us
consider some more manageable instance, such as the throwing of dice.
Every one who has played much with dice knows that double sixes are
sometimes thrown, and sometimes double aces. Such coincidences do not
happen once and only once; they occur again and again, and a great
number of trials will show that, though their recurrence has not the
regularity of cause and effect, it yet has a law of its own, namely--a
tendency to average regularity. In 10,000 throws there will be some
number of double sixes; and the greater the number of throws the more
closely will the average recurrence of double sixes, or double aces,
approximate to one in thirty-six. Such a law of average recurrence is
the basis of Probability. Chance being the fact of coincidence without
assignable cause, Probability is expectation based on the average
frequency of its happening.
Sec. 2. Probability is an ambiguous term. Usually, when we say that an
event is 'probable,' we mean that it is more likely than not to happen.
But, scientifically, an event is probable if our expectation of its
occurrence is less than certainty, as long as the event is not
impossible. Probability, thus conceived, is represented by a fraction.
Taking 1 to stand for certainty, and 0 for impossibility, probability
may be 999/1000, or 1/1000, or (generally) 1/_m_. The denominator
represents the number of times that an event happens, and the numerator
the number of times that it coincides with another event. In throwing a
die, the probability of ace turning up is expressed by putting the
number of throws for the denominator and the number of times that ace is
thrown for the numerator; and we may assume that the more trials we make
the nearer will the resulting fraction approximate to 1/6.
Instead of speaking of the 'throwing of the die' and its 'turning up
ace' as two events, the former is called 'the event' and the latter 'the
way of its happening.' And these expressions may easily be extended to
cover relations of distinct events; as when two men shoot at a mark and
we desire to represent the probability of both hitting the bull's eye
together, each shot may count as an event (denominator) and the
coincidence of 'bull's-eyes' as the way of its happening (numerator).
It is also common to speak of probability as a proportion. If the
fraction expressing the probability of ace being cast is 1/6, the
proportion of cases in which it happens is 1 to 5; or (as it is,
perhaps, still more commonly put) 'the chances are 5 to 1 against it.'
Sec. 3. As to the grounds of probability opinions differ. According to one
view the ground is subjective: probability depends, it is said, upon the
quantity of our Belief in the happening of a certain event, or in its
happening in a particular way. According to the other view the ground is
objective, and, in fact, is nothing else than experience, which is most
trustworthy when carefully expressed in statistics.
To the subjective view it may be objected, (a) that belief cannot by
itself be satisfactorily measured. No one will maintain that belief,
merely as a state of mind, always has a definite numerical value of
which one is conscious, as 1/100 or 1/10. Let anybody mix a number of
letters in a bag, knowing nothing of them except that one of them is X,
and then draw them one by one, endeavouring each time to estimate the
value of his belief that the next will be X; can he say that his belief
in the drawing of X next time regularly increases as the number of
letters left decreases?
If not, we see that (b) belief does not uniformly correspond with the
state of the facts. If in such a trial as proposed above, we really wish
to draw X, as when looking for something in a number of boxes, how
common it is, after a few failures, to feel quite hopeless and to say:
"Oh, of course it will be in the last." For belief is subject to hope
and fear, temperament, passion, and prejudice, and not merely to
rational considerations. And it is useless to appeal to 'the Wise Man,'
the purely rational judge of probability, unless he is producible. Or,
if it be said that belief is a short cut to the evaluation of
experience, because it is the resultant of all past experience, we may
reply that this is not true. For one striking experience, or two or
three recent ones, will immensely outweigh a great number of faint or
remote experiences. Moreover, the experience of two men may be
practically equal, whilst their beliefs upon any question greatly
differ. Any two Englishmen have about the same experience, personal and
ancestral, of the weather; yet their beliefs in the saw that 'if it rain
on St. Swithin's Day it will rain for forty days after,' may differ as
confident expectation and sheer scepticism. Upon which of these beliefs
shall we ground the probability of forty days' rain?
But (c) at any rate, if Probability is to be connected with Inductive
Logic, it must rest upon the same ground, namely--observation.
Induction, in any particular case, is not content with beliefs or
opinions, but aims at testing, verifying or correcting them by appealing
to the facts; and Probability has the same object and the same basis.
In some cases, indeed, the conditions of an event are supposed to be
mathematically predetermined, as in tossing a penny, throwing dice,
dealing cards. In throwing a die, the ways of happening are six; in
tossing a penny only two, head and tail: and we usually assume that the
odds with a die are fairly 5 to 1 against ace, whilst with a penny 'the
betting is even' on head or tail. Still, this assumption rests upon
another, that the die is perfectly fair, or that the head and tail of a
penny are exactly alike; and this is not true. With an ordinary die or
penny, a very great number of trials would, no doubt, give an average
approximating to 1/6 or 1/2; yet might always leave a certain excess one
way or the other, which would also become more definite as the trials
went on; thus showing that the die or penny did not satisfy the
mathematical hypothesis. Buffon is said to have tossed a coin 4040
times, obtaining 1992 heads and 2048 tails; a pupil of De Morgan tossed
4092 times, obtaining 2048 heads and 2044 tails.
There are other important cases in which probability is estimated and
numerically expressed, although statistical evidence directly bearing
upon the point in question cannot be obtained; as in betting upon a
race; or in the prices of stocks and shares, which are supposed to
represent the probability of their paying, or continuing to pay, a
certain rate of interest. But the judgment of experts in such matters is
certainly based upon experience; and great pains are taken to make the
evidence as definite as possible by comparing records of speed, or by
financial estimates; though something must still be allowed for reports
of the condition of horses, or of the prospects of war, harvests, etc.
However, where statistical evidence is obtainable, no one dreams of
estimating probability by the quantity of his belief. Insurance offices,
dealing with fire, shipwreck, death, accident, etc., prepare elaborate
statistics of these events, and regulate their rates accordingly. Apart
from statistics, at what rate ought the lives of men aged 40 to be
insured, in order to leave a profit of 5 per cent. upon L1000 payable at
each man's death? Is 'quantity of belief' a sufficient basis for doing
this sum?
Sec. 4. The ground of probability is experience, then, and, whenever
possible, statistics; which are a kind of induction. It has indeed been
urged that induction is itself based upon probability; that the
subtlety, complexity and secrecy of nature are such, that we are never
quite sure that we fully know even what we have observed; and that, as
for laws, the conditions of the universe at large may at any moment be
completely changed; so that all imperfect inductions, including the law
of causation itself, are only probable. But, clearly, this doctrine
turns upon another ambiguity in the word 'probable.' It may be used in
the sense of 'less than absolutely certain'; and such doubtless is the
condition of all human knowledge, in comparison with the comprehensive
intuition of arch-angels: or it may mean 'less than certain according
to _our_ standard of certainty,' that is, in comparison with the law of
causation and its derivatives.
We may suppose some one to object that "by this relative standard even
empirical laws cannot be called 'only probable' as long as we 'know no
exception to them'; for that is all that can be said for the boasted law
of causation; and that, accordingly, we can frame no fraction to
represent their probability. That 'all swans are white' was at one time,
from this point of view, not probable but certain; though we now know it
to be false. It would have been an indecorum to call it only probable as
long as no other-coloured swan had been discovered; not merely because
the quantity of belief amounted to certainty, but because the number of
events (seeing a swan) and the number of their happenings in a certain
way (being white) were equal, and therefore the evidence amounted to 1
or certainty." But, in fact, such an empirical law is only probable; and
the estimate of its probability must be based on the number of times
that similar laws have been found liable to exceptions. Albinism is of
frequent occurrence; and it is common to find closely allied varieties
of animals differing in colour. Had the evidence been duly weighed, it
could never have seemed more than probable that 'all swans are white.'
But what law, approaching the comprehensiveness of the law of causation,
presents any exceptions?
Supposing evidence to be ultimately nothing but accumulated experience,
the amount of it in favour of causation is incomparably greater than the
most that has ever been advanced to show the probability of any other
kind of event; and every relation of events which is shown to have the
marks of causation obtains the support of that incomparably greater body
of evidence. Hence the only way in which causation can be called
probable, for us, is by considering it as the upward limit (1) to which
the series of probabilities tends; as impossibility is the downward
limit (0). Induction, 'humanly speaking,' does not rest on probability;
but the probability of concrete events (not of mere mathematical
abstractions like the falling of absolutely true dice) rests on
induction and, therefore, on causation. The inductive evidence
underlying an estimate of probability may be of three kinds: (a) direct
statistics of the events in question; as when we find that, at the age
of 20, the average expectation of life is 39-40 years. This is an
empirical law, and, if we do not know the causes of any event, we must
be content with an empirical law. But (b) if we do know the causes of an
event, and the causes which may prevent its happening, and can estimate
the comparative frequency of their occurring, we may deduce the
probability that the effect (that is, the event in question) will occur.
Or (c) we may combine these two methods, verifying each by means of the
other. Now either the method (b) or (_a fortiori_) the method (c) (both
depending on causation) is more trustworthy than the method (a) by
itself.
But, further, a merely empirical statistical law will only be true as
long as the causes influencing the event remain the same. A die may be
found to turn ace once in six throws, on the average, in close
accordance with mathematical theory; but if we load it on that facet the
results will be very different. So it is with the expectation of life,
or fire, or shipwreck. The increased virulence of some epidemic such as
influenza, an outbreak of anarchic incendiarism, a moral epidemic of
over-loading ships, may deceive the hopes of insurance offices. Hence we
see, again, that probability depends upon causation, not causation upon
probability.
That uncertainty of an event which arises not from ignorance of the law
of its cause, but from our not knowing whether the cause itself does or
does not occur at any particular time, is Contingency.
Sec. 5. The nature of an average supposes deviations from it. Deviations
from an average, or "errors," are assumed to conform to the law (1) that
the greater errors are less frequent than the smaller, so that most
events approximate to the average; and (2) that errors have no "bias,"
but are equally frequent and equally great in both directions from the
mean, so that they are scattered symmetrically. Hence their distribution
may be expressed by some such figure as the following:
[Illustration: FIG. 11.]
Here _o_ is the average event, and _oy_ represents the number of average
events. Along _ox_, in either direction, deviations are measured. At _p_
the amount of error or deviation is _op_; and the number of such
deviations is represented by the line or ordinate _pa_. At _s_ the
deviation is _os_; and the number of such deviations is expressed by
_sb_. As the deviations grow greater, the number of them grows less. On
the other side of _o_, toward _-x_, at distances, _op'_, _os'_ (equal to
_op_, _os_) the lines _p'a'_, _s'b'_ represent the numbers of those
errors (equal to _pa_, _sb_).
If _o_ is the average height of the adult men of a nation, (say) 5 ft. 6
in., _s'_ and _s_ may stand for 5 ft. and 6 ft.; men of 4 ft. 6 in. lie
further toward _-x_, and men of 6 ft. 6 in. further toward _x_. There
are limits to the stature of human beings (or to any kind of animal or
plant) in both directions, because of the physical conditions of
generation and birth. With such events the curve _b'yb_ meets the
abscissa at some point in each direction; though where this occurs can
only be known by continually measuring dwarfs and giants. But in
throwing dice or tossing coins, whilst the average occurrence of ace is
once in six throws, and the average occurrence of 'tail' is once in two
tosses, there is no necessary limit to the sequences of ace or of 'tail'
that may occur in an infinite number of trials. To provide for such
cases the curve is drawn as if it never touched the abscissa.
That some such figure as that given above describes a frequent
characteristic of an average with the deviations from it, may be shown
in two ways: (1) By arranging the statistical results of any homogeneous
class of measurements; when it is often found that they do, in fact,
approximately conform to the figure; that very many events are near the
average; that errors are symmetrically distributed on either side, and
that the greater errors are the rarer. (2) By mathematical demonstration
based upon the supposition that each of the events in question is
influenced, more or less, by a number of unknown conditions common to
them all, and that these conditions are independent of one another. For
then, in rare cases, all the conditions will operate favourably in one
way, and the men will be tall; or in the opposite way, and the men will
be short; in more numerous cases, many of the conditions will operate in
one direction, and will be partially cancelled by a few opposing them;
whilst in still more cases opposed conditions will approximately balance
one another and produce the average event or something near it. The
results will then conform to the above figure.
From the above assumption it follows that the symmetrical curve
describes only a 'homogeneous class' of measurements; that is, a class
no portion of which is much influenced by conditions peculiar to itself.
If the class is not homogeneous, because some portion of it is subject
to _peculiar_ conditions, the curve will show a hump on one side or the
other. Suppose we are tabulating the ages at which Englishmen die who
have reached the age of 20, we may find that the greatest number die at
39 (19 years being the average expectation of life at 20) and that as
far as that age the curve upwards is regular, and that beyond the age of
39 it begins to descend regularly, but that on approaching 45 it bulges
out some way before resuming its regular descent--thus:
[Illustration: FIG. 12.]
Such a hump in the curve might be due to the presence of a considerable
body of teetotalers, whose longevity was increased by the peculiar
condition of abstaining from alcohol, and whose average age was 45, 6
years more than the average for common men.
Again, if the group we are measuring be subject to selection (such as
British soldiers, for which profession all volunteers below a certain
height--say, 5 ft. 5 in.--are rejected), the curve will fall steeply on
one side, thus:
[Illustration: FIG. 13.]
If, above a certain height, volunteers are also rejected, the curve will
fall abruptly on both sides. The average is supposed to be 5 ft. 8 in.
The distribution of events is described by 'some such curve' as that
given in Fig. 11; but different groups of events may present figures or
surfaces in which the slopes of the curves are very different, namely,
more or less steep; and if the curve is very steep, the figure runs into
a peak; whereas, if the curve is gradual, the figure is comparatively
flat. In the latter case, where the figure is flat, fewer events will
closely cluster about the average, and the deviations will be greater.
Suppose that we know nothing of a given event except that it belongs to
a certain class or series, what can we venture to infer of it from our
knowledge of the series? Let the event be the cephalic index of an
Englishman. The cephalic index is the breadth of a skull x 100 and
divided by the length of it; e.g. if a skull is 8 in. long and 6 in.
broad, (6x100)/8=75. We know that the average English skull has an index
of 78. The skull of the given individual, therefore, is more likely to
have that index than any other. Still, many skulls deviate from the
average, and we should like to know what is the probable error in this
case. The probable error is the measurement that divides the deviations
from the average in either direction into halves, so that there are as
many events having a greater deviation as there are events having a less
deviation. If, in Fig. 11 above, we have arranged the measurements of
the cephalic index of English adult males, and if at _o_ (the average or
mean) the index is 78, and if the line _pa_ divides the right side of
the fig. into halves, then _op_ is the probable error. If the
measurement at _p_ is 80, the probable error is 2. Similarly, on the
left hand, the probable error is _op'_, and the measurement at _p'_ is
76. We may infer, then, that the skull of the man before us is more
likely to have an index of 78 than any other; if any other, it is
equally likely to lie between 80 and 76, or to lie outside them; but as
the numbers rise above 80 to the right, or fall below 76 to the left, it
rapidly becomes less and less likely that they describe this skull.
In such cases as heights of men or skull measurements, where great
numbers of specimens exist, the average will be actually presented by
many of them; but if we take a small group, such as the measurements of
a college class, it may happen that the average height (say, 5 ft. 8
in.) is not the actual height of any one man. Even then there will
generally be a closer cluster of the actual heights about that number
than about any other. Still, with very few cases before us, it may be
better to take the median than the average. The median is that event on
either side of which there are equal numbers of deviations. One
advantage of this procedure is that it may save time and trouble. To
find approximately the average height of a class, arrange the men in
order of height, take the middle one and measure him. A further
advantage of this method is that it excludes the influence of
extraordinary deviations. Suppose we have seven cephalic indices, from
skeletons found in the same barrow, 75-1/2, 76, 78, 78, 79, 80-1/2, 86.
The average is 79; but this number is swollen unduly by the last
measurement; and the median, 78, is more fairly representative of the
series; that is to say, with a greater number of skulls the average
would probably have been nearer 78.
To make a single measurement of a phenomenon does not give one much
confidence. Another measurement is made; and then, if there is no
opportunity for more, one takes the mean or average of the two. But why?
For the result may certainly be worse than the first measurement.
Suppose that the events I am measuring are in fact fairly described by
Fig. II, although (at the outset) I know nothing about them; and that my
first measurement gives _p_, and my second _s_; the average of them is
worse than _p_. Still, being yet ignorant of the distribution of these
events, I do rightly in taking the average. For, as it happens, 3/4 of
the events lie to the left of _p_; so that if the first trial gives _p_,
then the average of _p_ and any subsequent trial that fell nearer than
(say) _s'_ on the opposite side, would be better than _p_; and since
deviations greater than _s'_ are rare, the chances are nearly 3 to 1
that the taking of an average will improve the observation. Only if the
first trial give _o_, or fall within a little more than 1/2 _p_ on
either side of _o_, will the chances be against any improvement by
trying again and taking an average. Since, therefore, we cannot know the
position of our first trial in relation to _o_, it is always prudent to
try again and take the average; and the more trials we can make and
average, the better is the result. The average of a number of
observations is called a "Reduced Observation."
We may have reason to believe that some of our measurements are better
than others because they have been taken by a better trained observer,
or by the same observer in a more deliberate way, or with better
instruments, and so forth. If so, such observations should be
'weighted,' or given more importance in our calculations; and a simple
way of doing this is to count them twice or oftener in taking the
average.
Sec. 6. These considerations have an important bearing upon the
interpretation of probabilities. The average probability for any
_general class_ or series of events cannot be confidently applied to any
_one instance_ or to any _special class_ of instances, since this one,
or this special class, may exhibit a striking error or deviation; it
may, in fact, be subject to special causes. Within the class whose
average is first taken, and which is described by general characters as
'a man,' or 'a die,' or 'a rifle shot,' there may be classes marked by
special characters and determined by special influences. Statistics
giving the average for 'mankind' may not be true of 'civilised men,' or
of any still smaller class such as 'Frenchmen.' Hence life-insurance
offices rely not merely on statistics of life and death in general, but
collect special evidence in respect of different ages and sexes, and
make further allowance for teetotalism, inherited disease, etc.
Similarly with individual cases: the average expectation for a class,
whether general or special, is only applicable to any particular case if
that case is adequately described by the class characters. In England,
for example, the average expectation of life for males at 20 years of
age is 39.40; but at 60 it is still 13.14, and at 73 it is 7.07; at 100
it's 1.61. Of men 20 years old those who live more or less than 39.40
years are deviations or errors; but there are a great many of them. To
insure the life of a single man at 20, in the expectation of his dying
at 60, would be a mere bet, if we had no special knowledge of him; the
safety of an insurance office lies in having so many clients that
opposite deviations cancel one another: the more clients the safer the
business. It is quite possible that a hundred men aged 20 should be
insured in one week and all of them die before 25; this would be
ruinous, if others did not live to be 80 or 90.
Not only in such a practical affair as insurance, but in matters purely
scientific, the minute and subtle peculiarities of individuals have
important consequences. Each man has a certain cast of mind, character,
physique, giving a distinctive turn to all his actions even when he
tries to be normal. In every employment this determines his Personal
Equation, or average deviation from the normal. The term Personal
Equation is used chiefly in connection with scientific observation, as
in Astronomy. Each observer is liable to be a little wrong, and this
error has to be allowed for and his observations corrected accordingly.
The use of the term 'expectation,' and of examples drawn from insurance
and gambling, may convey the notion that probability relates entirely to
future events; but if based on laws and causes, it can have no reference
to point of time. As long as conditions are the same, events will be the
same, whether we consider uniformities or averages. We may therefore
draw probable inferences concerning the past as well as the future,
subject to the same hypothesis, that the causes affecting the events in
question were the same and similarly combined. On the other hand, if we
know that conditions bearing on the subject of investigation, have
changed since statistics were collected, or were different at some time
previous to the collection of evidence, every probable inference based
on those statistics must be corrected by allowing for the altered
conditions, whether we desire to reason forwards or backwards in time.
Sec. 7. The rules for the combination of probabilities are as follows:
(1) If two events or causes do not concur, the probability of one or the
other occurring is the sum of the separate probabilities. A die cannot
turn up both ace and six; but the probability in favour of each is 1/6:
therefore, the probability in favour of one or the other is 1/3. Death
can hardly occur from both burning and drowning: if 1 in 1000 is burned
and 2 in 1000 are drowned, the probability of being burned or drowned is
3/1000.
(2) If two events are independent, having neither connection nor
repugnance, the probability of their concurring is found by multiplying
together the separate probabilities of each occurring. If in walking
down a certain street I meet A once in four times, and B once in three
times, I ought (by mere chance) to meet both once in twelve times: for
in twelve occasions I meet B four times; but once in four I meet A.
This is a very important rule in scientific investigation, since it
enables us to detect the presence of causation. For if the coincidence
of two events is more or less frequent than it would be if they were
entirely independent, there is either connection or repugnance between
them. If, e.g., in walking down the street I meet both A and B oftener
than once in twelve times, they may be engaged in similar business,
calling them from their offices at about the same hour. If I meet them
both less often than once in twelve times, they may belong to the same
office, where one acts as a substitute for the other. Similarly, if in a
multitude of throws a die turns six oftener than once in six times, it
is not a fair one: that is, there is a cause favouring the turning of
six. If of 20,000 people 500 see apparitions and 100 have friends
murdered, the chance of any man having both experiences is 1/8000; but
if each lives on the average 300,000 hours, the chance of both events
occurring in the same hour is 1/2400000000. If the two events occur in
the same hour oftener than this, there is more than a chance
coincidence.
The more minute a cause of connection or repugnance between events, the
longer the series of trials or instances necessary to bring out its
influence: the less a die is loaded, the more casts must be made before
it can be shown that a certain side tends to recur oftener than once in
six.
(3) The rule for calculating the probability of a dependent event is the
same as the above; for the concurrence of two independent events is
itself dependent upon each of them occurring. My meeting with both A and
B in the street is dependent on my walking there and on my meeting one
of them. Similarly, if A is sometimes a cause of B (though liable to be
frustrated), and B sometimes of C (C and B having no causes independent
of B and A respectively), the occurrence of C is dependent on that of B,
and that again on the occurrence of A. Hence we may state the rule: If
two events are dependent each on another, so that if one occur the
second may (or may not), and if the second a third; whilst the third
never occurs without the second, nor the second without the first; the
probability that if the first occur the third will, is found by
multiplying together the fractions expressing the probability that the
first is a mark of the second and the second of the third.
Upon this principle the value of hearsay evidence or tradition
deteriorates, and generally the cogency of any argument based upon the
combination of approximate generalisations dependent on one another or
"self-infirmative." If there are two witnesses, A and B, of whom A saw
an event, whilst B only heard A relate it (and is therefore dependent on
A), what credit is due to B's recital? Suppose the probability of each
man's being correct as to what he says he saw, or heard, is 3/4: then
(3/4 x 3/4 = 9/16) the probability that B's story is true is a little
more than 1/2. For if in 16 attestations A is wrong 4 times, B can only
be right in 3/4 of the remainder, or 9 times in 16. Again, if we have
the Approximate Generalisations, 'Most attempts to reduce wages are met
by strikes,' and 'Most strikes are successful,' and learn, on
statistical inquiry, that in every hundred attempts to reduce wages
there are 80 strikes, and that 70 p.c. of the strikes are successful,
then 56 p.c. of attempts to reduce wages are unsuccessful.
Of course this method of calculation cannot be quantitatively applied if
no statistics are obtainable, as in the testimony of witnesses; and even
if an average numerical value could be attached to the evidence of a
certain class of witnesses, it would be absurd to apply it to the
evidence of any particular member of the class without taking account of
his education, interest in the case, prejudice, or general capacity.
Still, the numerical illustration of the rapid deterioration of hearsay
evidence, when less than quite veracious, puts us on our guard against
rumour. To retail rumour may be as bad as to invent an original lie.
(4) If an event may coincide with two or more other independent events,
the probability that they will together be a sign of it, is found by
multiplying together the fractions representing the improbability that
each is a sign of it, and subtracting the product from unity.
This is the rule for estimating the cogency of circumstantial evidence
and analogical evidence; or, generally, for combining approximate
generalisations "self-corroboratively." If, for example, each of two
independent circumstances, A and B, indicates a probability of 6 to 1 in
favour of a certain event; taking 1 to represent certainty, 1-6/7 is the
improbability of the event, notwithstanding each circumstance. Then 1/7
x 1/7 = 1/49, the improbability of both proving it. Therefore the
probability of the event is 48 to 1. The matter may be plainer if put
thus: A's indication is right 6 times in 7, or 42 in 49; in the
remaining 7 times in 49, B's indication will be right 6 times.
Therefore, together they will be right 48 times in 49. If each of two
witnesses is truthful 6 times in 7, one or the other will be truthful 48
times in 49. But they will not be believed unless they agree; and in
the 42 cases of A being right, B will contradict him 6 times; so that
they only concur in being right 36 times. In the remaining 7 times in
which A is wrong, B will contradict him 6 times, and once they will both
be wrong. It does not follow that when both are wrong they will concur;
for they may tell very different stories and still contradict one
another.
If in an analogical argument there were 8 points of comparison, 5 for
and 3 against a certain inference, and the probability raised by each
point could be quantified, the total value of the evidence might be
estimated by doing similar sums for and against, and subtracting the
unfavourable from the favourable total.
When approximate generalisations that have not been precisely quantified
combine their evidence, the cogency of the argument increases in the
same way, though it cannot be made so definite. If it be true that most
poets are irritable, and also that most invalids are irritable, a still
greater proportion will be irritable of those who are both invalids and
poets.
On the whole, from the discussion of probabilities there emerge four
principal cautions as to their use: Not to make a pedantic parade of
numerical probability, where the numbers have not been ascertained; Not
to trust to our feeling of what is likely, if statistics can be
obtained; Not to apply an average probability to special classes or
individuals without inquiring whether they correspond to the average
type; and Not to trust to the empirical probability of events, if their
causes can be discovered and made the basis of reasoning which the
empirical probability may be used to verify.
The reader who wishes to pursue this subject further should read a work
to which the foregoing chapter is greatly indebted, Dr. Venn's _Logic of
Chance_.