_FALLACIES._
1. In works on Logic a chapter is generally devoted
to the discussion of Fallacies, that is, to the description
and classification of the different ways in which the rules
of Logic may be transgressed. The analogy of Probability
to Logic is sufficiently close to make it advisable to adopt
the same plan here. In describing his own opinions an
author is, of course, perpetually obliged to describe and
criticise those of others which he considers erroneous. But
some of the most widely spread errors find no supporters
worth mentioning, and exist only in vague popular misapprehension.
It will be found the best arrangement, therefore,
at the risk of occasional repetition, to collect a few of
the errors that occur most frequently, and as far as possible
to trace them to their sources; but it will hardly be worth
the trouble to attempt any regular system of arrangement
and classification. We shall mainly confine ourselves, in
accordance with the special province of this work, to problems
which involve questions of logical interest, or to those
which refer to the application of Probability to moral and
social science. We shall avoid the discussion of isolated
problems in games of chance and skill except when some
error of principle seems to be involved in them.
2. (I.) One of the most fertile sources of error and
confusion upon the subject has been already several times
alluded to, and in part discussed in a previous chapter.
This consists in choosing the class to which to refer an
event, and therefore judging of the rarity of the event and
the consequent improbability of foretelling it, _after it has
happened_, and then transferring the impressions we experience
to a supposed contemplation of the event beforehand.
The process in itself is perfectly legitimate (however unnecessary
it may be), since time does not in strictness enter
at all into questions of Probability. No error therefore
need arise in this way, if we were careful as to the class
which we thus selected; but such carefulness is often neglected.
An illustration may afford help here. A man once
pointed to a small target chalked upon a door, the target
having a bullet hole through the centre of it, and surprised
some spectators by declaring that he had fired that shot
from an old fowling-piece at a distance of a hundred yards,
His statement was true enough, but he suppressed a rather
important fact. The shot had really been aimed in a general
way at the barn-door, and had hit it; the target was afterwards
chalked round the spot where the bullet struck. A
deception analogous to this is, I think, often practised unconsciously
in other matters. We judge of events on a similar
principle, feeling and expressing surprise in an equally unreasonable
way, and deciding as to their occurrence on
grounds which are really merely a subsequent adjunct of our
own. Butler's remarks about 'the story of Cæsar,' discussed
already in the twelfth chapter, are of this character. He
selects a series of events from history, and then imagines a
person guessing them correctly who at the time had not the
history before him. As I have already pointed out, it is one
thing to be unlikely to guess an event rightly without
specific evidence; it is another and very different thing to
appreciate the truth of a story which is founded partly or
entirely upon evidence. But it is a great mistake to transfer
to one of these ways of viewing the matter the mental impressions
which properly belong to the other. It is like
drawing the target afterwards, and then being surprised to
find that the shot lies in the centre of it.
3. One aspect of this fallacy has been already discussed,
but it will serve to clear up difficulties which are
often felt upon the subject if we reexamine the question
under a somewhat more general form.
In the class of examples under discussion we are generally
presented with an individual which is not indeed definitely
referred to a class, but in regard to which we have no great
difficulty in choosing the appropriate class. Now suppose
we were contemplating such an event as the throwing of
sixes with a pair of dice four times running. Such a throw
would be termed a very unlikely event, as the odds against
its happening would be 36 × 36 × 36 × 36 - 1 to 1 or 1679615
to 1. The meaning of these phrases, as has been abundantly
pointed out, is simply that the event in question occurs very
rarely; that, stated with numerical accuracy, it occurs once in
1679616 times.
4. But now let us make the assumption that the
throw has actually occurred; let us put ourselves into the
position of contemplating sixes four times running when it is
known or reported that this throw has happened. The same
phrase, namely that the event is a very unlikely one, will
often be used in relation to it, but we shall find that this
phrase may be employed to indicate, on one occasion or
another, extremely different meanings.
(1) There is, firstly, the most correct meaning. The
event, it is true, has happened, and we know what it is, and
therefore, we have not really any occasion to resort to the
rules of Probability; but we can nevertheless conceive ourselves
as being in the position of a person who does not
know, and who has only Probability to appeal to. By calling
the chances 1679615 to 1 against the throw we then mean
to imply the fact, that inasmuch as such a throw occurs only
once in 1679616 times, our guess, were we to guess, would
be correct only once in the same number of times; provided,
that is, that it is a fair guess, based simply on these statistical
grounds.
5. (2) But there is a second and very different conception
sometimes introduced, especially when the event in
question is supposed to be known, not as above by the evidence
of our experience, but by the report of a witness. We
may then mean by the 'chances against the event' (as was
pointed out in Chapter XII.) not the proportional number of
times we should be right in guessing the event, but the
proportional number of times the witness will be right in
reporting it. The bases of our inference are here shifted
on to new ground. In the former case the statistics were
the throws and their respective frequency, now they are the
witnesses' statements and their respective truthfulness.
6. (3) But there is yet another meaning sometimes
intended to be conveyed when persons talk of the chances
against such an event as the throw in question. They may
mean--not, Here is an event, how often should I have
guessed it?--nor, Here is a report, how often will it be
correct?--but something different from either, namely, Here
is an event, how often will it be found to be produced by
some one particular kind of cause?
When, for example, a man hears of dice giving the same
throw several times running, and speaks of this as very
extraordinary, we shall often find that he is not merely
thinking of the improbability of his guess being right, or of
the report being true, but, that along with this, he is introducing
the question of the throw having been produced by
fair dice. There is, of course, no reason whatever why such
a question as this should not also be referred to Probability,
provided always that we could find the appropriate statistics
by which to judge. These statistics would be composed, not
of throws of the particular dice, nor of reports of the particular
witness, but of the occasions on which such a throw as
the one in question respectively had, and had not, been
produced fairly. The objection to entering upon this view
of the question would be that no such statistics are obtainable,
and that if they were, we should prefer to form our
opinion (on principles to be described in Chapter XVI.) from
the special circumstances of the case rather than from an
appeal to the average.
7. The reader will easily be able to supply examples
in illustration of the distinctions just given; we will briefly
examine but one. I hide a banknote in a certain book in a
large library, and leave the room. A person tells me that,
after I went out, a stranger came in, walked straight up to
that particular book, and took it away with him. Many
people on hearing this account would reply, How extremely
improbable! On analysing the phrase, I think we shall
find that certainly two, and possibly all three, of the above
meanings might be involved in this exclamation. (1) What
may be meant is this,--Assuming that the report is true,
and the stranger innocent, a rare event has occurred. Many
books might have been thus taken without that particular
one being selected. I should not therefore have expected
the event, and when it has happened I am surprised. Now
a man has a perfect right to be surprised, but he has no
logical right (so long as we confine ourselves to this view) to
make his surprise a ground for disbelieving the event. To
do this is to fall into the fallacy described at the commencement
of this chapter. The fact of my not having been likely
to have guessed a thing beforehand is no reason in itself for
doubting it when I am informed of it. (2) Or I may stop
short of the events reported, and apply the rules of Probability
to the report itself. If so, what I mean is that such a
story as this now before me is of a kind very generally false,
and that I cannot therefore attach much credit to it now.
(3) Or I may accept the truth of the report, but doubt the
fact of the stranger having taken the book at random. If
so, what I mean is, that of men who take books in the way
described, only a small proportion will be found to have
taken them really at random; the majority will do so because
they had by some means ascertained, or come to suspect,
what there was inside the book.
Each of the above three meanings is a possible and a
legitimate meaning. The only requisite is that we should be
careful to ascertain which of them is present to the mind, so
as to select the appropriate statistics. The first makes in
itself the most legitimate use of Probability; the drawback
being that at the time in question the functions of Probability
are superseded by the event being otherwise known.
The second or third, therefore, is the more likely meaning to
be present to the mind, for in these cases Probability, if it
could be practically made use of, would, at the time in
question, be a means of drawing really important inferences.
The drawbacks are the difficulty of finding such statistics,
and the extreme disturbing influence upon these statistics of
the circumstances of the special case.
8. (II.) Closely connected with the tendency just
mentioned is that which prompts us to confound a true
chance selection with one which is more or less picked.
When we are dealing with familiar objects in a concrete
way, especially when the greater rarity corresponds to superiority
of quality, almost every one has learnt to recognize
the distinction. No one, for instance, on observing a fine
body of troops in a foreign town, but would be prompted to
ask whether they came from an average regiment or from
one that was picked. When however the distinction refers
to unfamiliar objects, and especially when only comparative
rarity seems to be involved, the fallacy may assume a rather
subtle and misleading form, and seems to deserve special
notice by the consideration of a few examples.
Sometimes the result is not so much an actual fallacy as
a slight misreckoning of the order of probability of the event
under consideration. For instance, in the Pyramid question,
we saw that it made some difference whether we considered
that π alone was to be taken into account or whether we
put this constant into a class with a small number of other
similar ones. In deciding, however, whether or not there is
anything remarkable in the actual falling short of the
representation of the number 7 in the evaluation of π
(v. p. 248) the whole question turns upon considerations of
this kind. The only enquiry raised is whether there is anything
remarkable in this departure from the mean, and the
answer depends upon whether we suppose that we are referring
to a predetermined digit, or to whatever digit of the
ten happens to be most above or below the average. Or,
take the case raised by Cournot (_Exposition de la Théorie
des Chances_, §§102, 114), that a certain deviation from the
mean in the case of Departmental returns of the proportion
between male and female births is significant and indicative
of a difference in kind, provided that we select at random a
single French Department; but that the same deviation may
be accidental if it is the maximum of the respective returns
for several Departments.[1] The answer may be given one
way or the other according as we bear this consideration in
mind.
9. We are peculiarly liable to be misled in this way
when we are endeavouring to determine the cause of some
phenomenon, by mere statistics, in entire ignorance as to the
direction in which the cause should be expected. In such
cases an ingenious person who chooses to look about over a
large field can never fail to hit upon an explanation which is
plausible in the sense that it fits in with the hitherto observed
facts. With a tithe of the trouble which Mr Piazzi
Smyth expended upon the measurement of the great pyramid,
I think I would undertake to find plausible intimations of
several of the important constants and standards which he
discovered there, in the dimensions of the desk at which I am
writing. The oddest instance of this sort of conclusion is
perhaps to be found in the researches of a writer who has
discovered[2] that there is a connection of a striking kind
between the respective successes of the Oxford and the
Cambridge boat in the annual race, and the greater and less
frequency of sun-spots.
Of course our usual practical resource in such cases is to
make appeal to our previous knowledge of the subject in
question, which enables us to reject as absurd a great number
of hypotheses which can nevertheless make a fair show when
they are allowed to rest upon a limited amount of adroitly
selected instances. But it must be remembered that if any
theory chooses to appeal to statistics, to statistics it must be
suffered to go for judgment. Even the boat race theory
could be established (if sound) on this ground alone. That
is, if it really could be shown that experience in the long run
confirmed the preponderance of successes on one side or the
other according to the relative frequency of the sun-spots,
we should have to accept the fact that the two classes of
events were not really independent. One of the two, whichever
it may be, must be suspected of causing or influencing
the other; or both must be caused or influenced by some
common circumstances.
10. (III.) The fallacy described at the commencement
of this chapter arose from determining to judge of an observed
or reported event by the rules of Probability, but
employing a wrong set of statistics in the process of judging.
Another fallacy, closely connected with this, arises from the
practice of taking some only of the characteristics of such an
event, and arbitrarily confining to these the appeal to Probability.
Suppose I toss up twelve pence and find that eleven
of them give heads. Many persons on witnessing such an occurrence
would experience a feeling which they would express by
the remark, How near that was to getting all heads! And if
any thing very important were staked on the throw they
would be much excited at the occurrence. But in what
sense were we near to twelve? There is a not uncommon
error, I apprehend, which consists in unconsciously regarding
the eleven heads as a thing which is already somehow
secured, so that one might as it were keep them, and then
take our chance for securing the remaining one. The eleven
are mentally set aside, looked upon as certain (for they have
already happened), and we then introduce the notion of
chance merely for the twelfth. But this twelfth, having
also happened, has no better claim to such a distinction than
any of the others. If we will introduce the notion of chance
in the case of the one that gave tail we must do the same in
the case of all the others as well. In other words, if the
tosser be dissatisfied at the appearance of the one tail, and
wish to cancel it and try his luck again, he must toss up the
whole lot of pence again fairly together. In this case, of
course, so far from his having a better prospect for the next
throw he may think himself in very good luck if he makes
again as good a throw as the one he rejected. What he is
doing is confounding this case with that in which the throws
are really _successive_. If eleven heads have been tossed up in
turn, we are of course within an even chance of getting a
twelfth; but the circumstances are quite different in the
instance proposed.
11. In the above example the error is transparent.
But in forming a judgment upon matters of greater complexity
than dice and pence, especially in the case of what are
called 'narrow escapes,' a mistake of an analogous kind is, I
apprehend, far from uncommon. A person, for example, who
has just experienced a narrow escape will often be filled
with surprise and anxiety amounting almost to terror. The
event being past, these feelings are, at the time, in strictness
inappropriate. If, as is quite possible, they are merely instinctive,
or the result of association, they do not fall within
the province of any kind of Logic. If, however, as seems
more likely, they partially arise from a supposed transference
of ourselves into that point of past time at which the event
was just about to happen, and the production by imagination
of the feelings we should then expect to experience,
this process partakes of the nature of an inference, and can
be right or wrong. In other words, the alarm may be proportionate
or disproportionate to the amount of danger that
might fairly have been reckoned upon in such a hypothetical
anticipation. If the supposed transfer were completely
carried out, there would be no fallacy; but it is often very
incompletely done, some of the component parts of the event
being supposed to be determined or 'arranged' (to use a
sporting phrase) in the form in which we now know that
they actually have happened, and only the remaining ones
being fairly contemplated as future chances.
A man, for example, is out with a friend, whose rifle goes
off by accident, and the bullet passes through his hat. He
trembles with anxiety at thinking what might have happened,
and perhaps remarks, 'How very near I was to being
killed!' Now we may safely assume that he means something
more than that a shot passed very close to him. He
has some vague idea that, as he would probably say, 'his
chance of being killed then was very great.' His surprise
and terror may be in great part physical and instinctive,
arising simply from the knowledge that the shot had passed
very near him. But his mental state may be analysed, and
we shall then most likely find, at bottom, a fallacy of the
kind described above. To speak or think of chance in connection
with the incident, is to refer the particular incident
to a class of incidents of a similar character, and then to consider
the comparative frequency with which the contemplated
result ensues. Now the series which we may suppose
to be most naturally selected in this case is one composed of
shooting excursions with his friend; up to this point the
proceedings are assumed to be designed, beyond it only,
in the subsequent event, was there accident. Once in a
thousand times perhaps on such occasions the gun will go
off accidentally; one in a thousand only of those discharges
will be directed near his friend's head. If we will make the
accident a matter of Probability, we ought by rights in this
way (to adopt the language of the first example), to 'toss up
again' fairly. But we do not do this; we seem to assume for
certain that the shot goes within an inch of our heads, detach
that from the notion of chance at all, and then begin to
introduce this notion again for possible deflections from that
saving inch.
12. (IV.) We will now notice a fallacy connected with
the subjects of betting and gambling. Many or most of the
popular misapprehensions on this subject imply such utter
ignorance and confusion as to the foundations of the science
that it would be needless to discuss them here. The following
however is of a far more plausible kind, and has been a
source of perplexity to persons of considerable acuteness.
The case, put into the simplest form, is as follows.[3]
Suppose that a person A is playing against B, B being
either another individual or a group of individuals, say a
gambling bank. They begin by tossing for a shilling, and
A maintains that he is in possession of a device which will
insure his winning. If he does win on the first occasion he
has clearly gained his point so far. If he loses, he stakes
next time two shillings instead of one. The result of course
is that if he wins on the second occasion he replaces his
former loss, and is left with one shilling profit as well. So
he goes on, doubling his stake after every loss, with the
obvious result that on the first occasion of success he makes
good all his previous losses, and is left with a shilling over.
But such an occasion must come sooner or later, by the
assumptions of chance on which the game is founded. Hence
it follows that he can insure, sooner or later, being left a
final winner. Moreover he may win to any amount; firstly
from the obvious consideration that he might make his
initial stake as large as he pleased, a hundred pounds, for
instance, instead of a shilling; and secondly, because what
he has done once he may do again. He may put his shilling
by, and have a second spell of play, long or short as the case
may be, with the same termination to it. Accordingly by
mere persistency he may accumulate any sum of money he
pleases, in apparent defiance of all that is meant by luck.
13. I have classed this opinion among fallacies, as the
present is the most convenient opportunity of discussing it,
though in strictness it should rather be termed a paradox,
since the conclusion is perfectly sound. The only fallacy
consists in regarding such a way of obtaining the result as
mysterious. On the contrary, there is nothing more easy
than to insure ultimate success under the given conditions.
The point is worth enquiry, from the principles it involves,
and because the answers commonly given do not quite meet
the difficulty. It is sometimes urged, for instance, that no
bank would or does allow the speculator to choose at will the
amount of his stake, but puts a limit to the amount for
which it will consent to play. This is quite true, but is of
course no answer to the hypothetical enquiry before us, which
assumes that such a state of things _is_ allowed. Again, it has
been urged that the possibility in question turns entirely
upon the fact that credit must be supposed to be given, for
otherwise the fortune of the player may not hold out until
his turn of luck arrives:--that, in fact, sooner or later, if he
goes on long enough, his fortune will not hold out long
enough, and all his gains will be swept away. It is quite
true that credit is a _condition_ of success, but it is in no sense
the cause. We may suppose both parties to agree at the
outset that there shall be no payments until the game be
ended, A having the right to decide when it shall be considered
to be ended. It still remains true that whereas in
ordinary gambling, i.e. with fixed or haphazard stakes, A could
not ensure winning eventually to any extent, he can
do so if he adopt such a scheme as the one in question. And
this is the state of things which seems to call for explanation.
14. What causes perplexity here is the supposed fact
that in some mysterious way certainty has been conjured out
of uncertainty; that in a game where the detailed events are
utterly inscrutable, and where the average, by supposition,
shows no preference for either side, one party is nevertheless
succeeding somehow in steadily drawing the luck his own
way. It looks as if it were a parallel case with that of a
man who should succeed by some device in permanently
securing more than half of the tosses with a penny which
was nevertheless to be regarded as a perfectly fair one.
This is quite a mistake. The real fact is that A does
not expose his gains to chance at all; all that he so exposes
is the number of times he has to wait until he gains. Put
such a case as this. I offer to give a man any sum of money
he chooses to mention provided he will at once give it back
again to me with one pound more. It does not need much
acuteness to see that it is a matter of indifference to me
whether he chooses to mention one pound, or ten, or a
hundred. Now suppose that instead of leaving it to his
choice which of these sums is to be selected each time, the
two parties agree to leave it to chance. Let them, for
instance, draw a number out of a bag each time, and let that
be the sum which A gives to B under the prescribed conditions.
The case is not altered. A still gains his pound each
time, for the introduction of the element of chance has not
in any way touched this. All that it does is to make this
pound the result of an uncertain subtraction, sometimes 10
_minus_ 9, sometimes 50 _minus_ 49, and so on. It is these
numbers only, not their difference, which he submits to luck,
and this is of no consequence whatever.
To suggest to any individual or company that they
should consent to go on playing upon such terms as these
would be too barefaced a proposal. And yet the case in
question is identical in principle, and almost identical in
form, with this. To offer to give a man any sum he likes to
name provided he gives you back again that same sum _plus_
one, and to offer him any number of terms he pleases of the
series 1, 2, 4, 8, 16, &c., provided you have the next term of
the set, are equivalent. The only difference is that in the
latter case the result is attained with somewhat more of
arithmetical parade. Similarly equivalent are the processes
in case we prefer to leave it to chance, instead of to choice,
to decide what sum or what number of terms shall be fixed
upon. This latter is what is really done in the case in
question. A man who consents to go on doubling his stake
every time he wins, is leaving nothing else to chance than
the determination of the particular number of terms of such
a geometrical series which shall be allowed to pass before he
stops.
15. It may be added that there is no special virtue in
the particular series in question, viz. that in accordance with
which the stake is doubled each time. All that is needed is
that the last term of the series should more than balance all
the preceding ones. Any other series which increased faster
than this geometrical one, would answer the purpose as well
or better. Nor is it necessary, again, that the game should
be an even or 'fair' one. Chance, be it remembered, affects
nothing here but the number of terms to which the series
attains on each occasion, its final result being always arithmetically
fixed. When a penny is tossed up it is only on
one of every two occasions that the series runs to more than
two terms, and so his fixed gains come in pretty regularly.
But unless he was playing for a limited time only, it would
not affect him if the series ran to two hundred terms; it
would merely take him somewhat longer to win his stakes.
A man might safely, for instance, continue to lay an _even_
bet that he would get the single prize in a lottery of a thousand
tickets, provided he thus doubled, or more than doubled,
his stake each time, and unlimited credit was given.
16. So regarded, the problem is simple enough, but
there are two points in it to which attention may conveniently
be directed.
In the first place, it serves very pointedly to remind us of
the distinction between a series of events (in this case the
tosses of the penny) which really are subjects of chance, and
our conduct founded upon these events, which may or may
not be so subject.[4] It is quite possible that this latter may
be so contrived as to be in many respects a matter of absolute
certainty,--a consideration, I presume, familiar enough
to professional betting men. Why is the ordinary way of
betting on the throws of a penny fair to both parties? Because
a 'fair' series is 'fairly' treated. The heads and tails
occur at random, but on an average equally often, and the
stakes are either fixed or also arranged at random. If a man
backs heads every time for the same amount, he will of
course in the long run neither win nor lose. Neither will he
if he varies the stake every time, provided he does not vary
it in such a way as to make its amount dependent on the
fact of his having won or lost the time before. But he
may, if he pleases, and the other party consents, so arrange
his stakes (as in the case in question) that Chance, if one
might so express it, does not get a fair chance. Here the
human elements of choice and design have been so brought
to bear upon a series of events which, regarded by themselves,
exhibit nothing but the physical characteristics of
chance, that the latter elements disappear, and we get a
result which is arithmetically certain. Other analogous
instances might be suggested, but the one before us has the
merit of most ingeniously disguising the actual process.
17. The meaning of the remark just made will be
better seen by a comparison with the following case. It has
been attempted[5] to explain the preponderance of male births
over female by assuming that the chances of the two are
equal, but that the general desire to have a male heir tends
to induce many unions to persist until the occurrence of this
event, and no longer. It is supposed that in this way there
would be a slight preponderance of families which consisted
of one son only, or of two sons and one daughter, and so forth.
This is quite fallacious (as had been noticed by Laplace,
in his _Essai_); and there could not be a better instance
chosen than this to show just what we can do and what we
cannot do in the way of altering the luck in a real chance-succession
of events. To suppose that the number of actual
births could be influenced in the way in question is exactly
the same thing as to suppose that a number of gamblers
could increase the ratio of heads to tails, to something over
one-half, by each handing the coin to his neighbour as soon
as he had thrown a head: that they have only to leave off as
soon as head has appeared; an absurdity which we need not
pause to explain at this stage. The essential point about
the 'Martingale' is that, whereas the occurrence of the
events on which the stakes are laid is unaffected, the stakes
themselves can be so adjusted as to make the luck swing one
way.
18. In the second place, this example brings before us
what has had to be so often mentioned already, namely, that
the series of Probability are in strictness supposed to be
interminable. If therefore we allow either party to call
upon us to stop, especially at a point which just happens to
suit him, we may get results decidedly opposed to the
integrity of the theory. In the case before us it is a necessary
stipulation for A that he may be allowed to leave off
when he wishes, that is at one of the points at which the
throw is in his favour. Without this stipulation he may be
left a loser to any amount.
Introduce the supposition that one party may arbitrarily
call for a stoppage when it suits him and refuse to permit it
sooner, and almost any system of what would be otherwise fair
play may be converted into a very one-sided arrangement.
Indeed, in the case in question, A need not adopt this device
of doubling the stakes every time he loses. He may play
with a fixed stake, and nevertheless insure that _one_ party
shall win any assigned sum, assuming that the game is even
and that he is permitted to play on credit.
19. (V.) A common mistake is to assume that a very
unlikely thing will not happen at all. It is a mistake which,
when thus stated in words, is too obvious to be committed,
for the meaning of an unlikely thing is one that happens at
rare intervals; if it were not assumed that the event would
happen sometimes it would not be called unlikely, but impossible.
This is an error which could scarcely occur except
in vague popular misapprehension, and is so abundantly refuted
in works on Probability, that it need only be touched
upon briefly here. It follows of course, from our definition
of Probability, that to speak of a very rare combination of
events as one that is 'sure never to happen,' is to use language
incorrectly. Such a phrase may pass current as a
loose popular exaggeration, but in strictness it involves a
contradiction. The truth about such rare events cannot be
better described than in the following quotation from De Morgan:[6]--
"It is said that no person ever _does_ arrive at such extremely
improbable cases as the one just cited [drawing the
same ball five times running out of a bag containing twenty
balls]. That a given individual should never throw an ace
twelve times running on a single die, is by far the most
likely; indeed, so remote are the chances of such an event
in any twelve trials (more than 2,000,000,000 to 1 against it)
that it is unlikely the experience of any given country, in
any given century, should furnish it. But let us stop for a
moment, and ask ourselves to what this argument applies.
A person who rarely touches dice will hardly believe that
doublets sometimes occur three times running; one who handles
them frequently knows that such is sometimes the fact.
Every very practised user of those implements has seen still
rarer sequences. Now suppose that a society of persons had
thrown the dice so often as to secure a run of six aces observed
and recorded, the preceding argument would still be
used against twelve. And if another society had practised
long enough to see twelve aces following each other, they
might still employ the same method of doubting as to a run
of twenty-four; and so on, _ad infinitum_. The power of imagining
cases which contain long combinations so much exceeds
that of exhibiting and arranging them, that it is easy
to assign a telegraph which should make a separate signal
for every grain of sand in a globe as large as the visible universe,
upon the hypothesis of the most space-penetrating
astronomer. The fallacy of the preceding objection lies in
supposing events in number beyond our experience, composed
entirely of sequences such as fall within our experience. It
makes the past necessarily contain the whole, as to the quality
of its components; and judges by samples. Now the
least cautious buyer of grain requires to examine a handful
before he judges of a bushel, and a bushel before he judges
of a load. But relatively to such enormous numbers of combinations
as are frequently proposed, our experience does not
deserve the title of a handful as compared with a bushel, or
even of a single grain."
20. The origin of this inveterate mistake is not difficult
to be accounted for. It arises, no doubt, from the exigencies
of our practical life. No man can bear in mind
every contingency to which he may be exposed. If therefore
we are ever to do anything at all in the world, a large number
of the rarer contingencies must be left entirely out of
account. And the necessity of this oblivion is strengthened
by the shortness of our life. Mathematically speaking, it
would be said to be certain that any one who lives long
enough will be bitten by a mad dog, for the event is not an
impossible, but only an improbable one, and must therefore
come to pass in time. But this and an indefinite number
of other disagreeable contingencies have on most occasions
to be entirely ignored in practice, and thence they come
almost necessarily to drop equally out of our thought and
expectation. And when the event is one in itself of no importance,
like a rare throw of the dice, a great effort of
imagination may be required, on the part of persons not accustomed
to abstract mathematical calculation, to enable
them to realize the throw as being even possible.
Attempts have sometimes been made to estimate what
extremity of unlikelihood ought to be considered as equivalent
to this practical zero point of belief. In so far as
such attempts are carried out by logicians, or by those who
are unwilling to resort to mathematical valuation of chances,
they must be regarded as merely a special form of the modal
difficulties discussed in the last chapter, and need not therefore
be reconsidered here; but a word or two may be added
concerning the views of some who have looked at the matter
from the mathematician's point of view.
The principal of these is perhaps Buffon. He has arrived
at the estimate (_Arithmétique Morale_ §VIII.) that this
practical zero is equivalent to a chance of 1/10,000. The
grounds for selecting this fraction are found in the fact that,
according to the tables of mortality accessible to him, it
represents the chance of a man of 56 dying in the course of
the next day. But since no man under common circumstances
takes the chance into the slightest consideration, it
follows that it is practically estimated as having no value.
It is obvious that this result is almost entirely arbitrary,
and in fact his reasons cannot be regarded as anything more
than a slender justification from experience for adopting a
conveniently simple fraction; a justification however which
would apparently have been equally available in the case of
any other fractions lying within wide limits of the one
selected.[7]
21. There is one particular form of this error, which,
from the importance occasionally attached to it, deserves
perhaps more special examination. As stated above, there
can be no doubt that, however unlikely an event may be, if
we (loosely speaking) vary the circumstances sufficiently, or
if, in other words, we keep on trying long enough, we shall
meet with such an event at last. If we toss up a pair of
dice a few times we shall get doublets; if we try longer with
three we shall get triplets, and so on. However unusual the
event may be, even were it sixes a thousand times running,
it will come some time or other if we have only patience and
vitality enough. Now apply this result to the letters of the
alphabet. Suppose that one letter at a time is drawn from
a bag which contains them all, and is then replaced. If the
letters were written down one after another as they occurred,
it would commonly be expected that they would be found to
make mere nonsense, and would never arrange themselves
into the words of any language known to men. No more
they would in general, but it is a commonly accepted result
of the theory, and one which we may assume the reader to
be ready to admit without further discussion, that, if the
process were continued long enough, words making sense
would appear; nay more, that any book we chose to mention,--Milton's
_Paradise Lost_ or the plays of Shakespeare,
for example,--would be produced in this way at last. It
would take more days than we have space in this volume to
represent in figures, to make tolerably certain of obtaining
the former of these works by thus drawing letters out of a
bag, but the desired result would be obtained at length.[8]
Now many people have not unnaturally thought it derogatory
to genius to suggest that its productions could
have also been obtained by chance, whilst others have gone
on to argue, If this be the case, might not the world itself in
this manner have been produced by chance?
22. We will begin with the comparatively simple, determinate,
and intelligible problem of the possible production
of the works of a great human genius by chance. With
regard to this possibility, it may be a consolation to some
timid minds to be reminded that the power of producing the
works of a Shakespeare, _in time_, is not confined to consummate
genius and to mere chance. There is a third alternative,
viz. that of purely mechanical procedure. Any one,
down almost to an idiot, might do it, if he took sufficient
time about the task. For suppose that the required number
of letters were procured and arranged, not by chance, but
designedly, and according to rules suggested by the theory
of permutations: the letters of the alphabet and the number
of them to be employed being finite, every order in which
they could occur would come in its due turn, and therefore
every thing which can be expressed in language would be
arrived at some time or other.
There is really nothing that need shock any one in such
a result. Its possibility arises from the following cause.
The number of letters, and therefore of words, at our disposal
is limited; whatever therefore we may desire to express
in language necessarily becomes subject to corresponding
limitation. The possible variations of thought are literally
infinite, so are those of spoken language (by intonation of
the voice, &c.); but when we come to words there is a limitation,
the nature of which is distinctly conceivable by the
mind, though the restriction is one that in practice will
never be appreciable, owing to the fact that the number of
combinations which may be produced is so enormous as to
surpass all power of the imagination to realize.[9] The answer
therefore is plain, and it is one that will apply to many other
cases as well, that to put a finite limit upon the number of
ways in which a thing can be done, is to determine that any
one who is able and willing to try long enough shall succeed
in doing it. If a great genius condescends to perform it
under these circumstances, he must submit to the possibility
of having his claims rivalled or disputed by the chance-man
and idiot. If Shakespeare were limited to the use of eight
or nine assigned words, the time within which the latter
agents might claim equality with him would not be very
great. As it is, having had the range of the English language
at his disposal, his reputation is not in danger of being
assailed by any such methods.
23. The case of the possible production of the world
by chance leads us into an altogether different region of discussion.
We are not here dealing with figures the nature
and use of which are within the fair powers of the understanding,
however the imagination may break down in attempting
to realize the smallest fraction of their full significance.
The understanding itself is wandering out of its
proper province, for the conditions of the problem cannot be
assigned. When we draw letters out of a bag we know very
well what we are doing; but what is really meant by producing
a world by chance? By analogy of the former case,
we may assume that some kind of agent is presupposed;--perhaps
therefore the following supposition is less absurd
than any other. Imagine some being, not a Creator but a
sort of Demiurgus, who has had a quantity of materials put
into his hands, and he assigns them their collocations and
their laws of action, blindly and at haphazard: what are the
odds that such a world as we actually experience should have
been brought about in this way?
If it were worth while seriously to set about answering
such a question, and if some one would furnish us with the
number of the letters of such an alphabet, and the length of
the work to be written with them, we could proceed to indicate
the result. But so much as this may surely be affirmed
about it;--that, far from merely finding the length of this
small volume insufficient for containing the figures in which
the adverse odds would be given, all the paper which the
world has hitherto produced would be used up before we had
got far on our way in writing them down.
24. The most seductive form in which the difficulty
about the occurrence of very rare events generally presents
itself is probably this. 'You admit (some persons will be
disposed to say) that such an event may sometimes happen;
nay, that it does sometimes happen in the infinite course of
time. How then am I to know that _this_ occasion is not one
of these possible occurrences?' To this, one answer only can
be given,--the same which must always be given where
statistics and probability are concerned,--'The present _may_
be such an occasion, but it is inconceivably unlikely that it
should be one. Amongst countless billions of times in which
you, and such as you, urge this, one person only will be
justified; and it is not likely that you are that one, or that
this is that occasion.'
25. There is another form of this practical inability to
distinguish between one high number and another in the
estimation of chances, which deserves passing notice from its
importance in arguments about heredity. People will often
urge an objection to the doctrine that qualities, mental and
bodily, are transmitted from the parents to the offspring, on
the ground that there are a multitude of instances to the
contrary, in fact a great majority of such instances. To
raise this objection implies an utter want of appreciation of
the very great odds which possibly may exist, and which the
argument in support of heredity implies _do_ exist against any
given person being distinguished for intellectual or other
eminence. This is doubtless partly a matter of definition,
depending upon the degree of rarity which we consider to be
implied by eminence; but taking any reasonable sense of the
term, we shall readily see that a very great proportion of
failures may still leave an enormous preponderance of evidence
in favour of the heredity doctrine. Take, for instance,
that degree of eminence which is implied by being one of
four thousand. This is a considerable distinction, though,
since there are about two thousand such persons to be found
amongst the total adult male population of Great Britain, it
is far from implying any conspicuous genius. Now suppose
that in examining the cases of a large number of the children
of such persons, we had found that 199 out of 200 of
them failed to reach the same distinction. Many persons
would conclude that this was pretty conclusive evidence
against any hereditary transmission. To be able to adduce
only one favourable, as against 199 hostile instances, would
to them represent the entire break-down of any such theory.
The error, of course, is obvious enough, and one which, with
the figures thus before him, hardly any one could fail to
avoid. But if one may judge from common conversation
and other such sources of information, it is found in practice
exceedingly difficult adequately to retain the conviction that
even though only one in 200 instances were favourable,
this would represent odds of about 20 to 1 in favour of the
theory. If hereditary transmission did not prevail, only one
in 4000 sons would thus rival their fathers; but we find
actually, let us say (we are of course taking imaginary proportions
here), that one in 200 does. Hence, if the statistics
are large enough to be satisfactory, there has been some
influence at work which has improved the chances of mere
coincidence in the ratio of 20 to 1. We are in fact so little
able to realise the meaning of very large numbers,--that is,
to retain the _ratios_ in the mind, where large numbers are
concerned,--that unless we repeatedly check ourselves by
arithmetical considerations we are too apt to treat and estimate
all beyond certain limits as equally vast and vague.
26. (VI.) In discussing the nature of the connexion
between Probability and Induction, we examined the claims
of a rule commonly given for inferring the probability that
an event which had been repeatedly observed would recur
again. I endeavoured to show that all attempts to obtain
and prove such a rule were necessarily futile; if these reasons
were conclusive the employment of such a rule must of
course be regarded as fallacious. A few examples may conveniently
be added here, tending to show how instead of
there being merely a single rule of succession we might better
divide the possible forms into three classes.
(1) In some cases when a thing has been observed to
happen several times it becomes in consequence _more_ likely
that the thing should happen again. This agrees with the
ordinary form of the rule, and is probably the case of most
frequent occurrence. The necessary vagueness of expression
when we talk of the 'happening of a thing' makes it quite
impossible to tolerate the rule in this general form, but if we
specialize it a little we shall find it assume a more familiar
shape. If, for example, we have observed two or more properties
to be frequently associated together in a succession of
individuals, we shall conclude with some force that they will
be found to be so connected in future. The strength of our
conviction however will depend not merely on the number
of observed coincidences, but on far more complicated considerations;
for a discussion of which the reader must be
referred to regular treatises on Inductive evidence. Or again,
if we have observed one of two events succeed the other
several times, the occurrence of the former will excite in
most cases some degree of expectation of the latter. As
before, however, the degree of our expectation is not to be
assigned by any simple formula; it will depend in part upon
the supposed intimacy with which the events are connected.
To attempt to lay down definite rules upon the subject
would lead to a discussion upon laws of causation, and the
circumstances under which their existence may be inferred,
and therefore any further consideration of the matter must
be abandoned here.
27. (2) Or, secondly, the past recurrence may in itself
give no valid grounds for inference about the future;
this is the case which most properly belongs to Probability.[10]
That it does so belong will be easily seen if we bear in mind
the fundamental conception of the science. We are there
introduced to a series,--for purposes of inference an indefinitely
extended series,--of terms, about the details of which,
information, except on certain points, is not given; our knowledge
being confined to the statistical fact, that, say, one in
ten of them has some attribute which we will call X. Suppose
now that five of these terms in succession have been X,
what hint does this give about the sixth being also an X?
Clearly none at all; this past fact tells us nothing; the formula
for our inference is still precisely what it was before,
that one in ten being X it is one to nine that the next term
is X. And however many terms in succession had been of
one kind, precisely the same formula would still be given.
28. The way in which events will justify the answer
given by this formula is often misunderstood. For the
benefit therefore of those unacquainted with some of the
conceptions familiar to mathematicians, a few words of explanation
may be added. Suppose then that we have had
X twelve times in succession. This is clearly an anomalous
state of things. To suppose anything like this continuing
to occur would be obviously in opposition to the statistics,
which assert that in the long run only one in ten is X.
But how is this anomaly got over? In other words, how
do we obviate the conclusion that X's must occur more
frequently than once in ten times, after such a long succession
of them as we have now had? Many people seem to
believe that there must be a diminution of X's afterwards
to counterbalance their past preponderance. This however
would be quite a mistake; the proportion in which they
occur in future must remain the same throughout; it cannot
be altered if we are to adhere to our statistical formula.
The fact is that the rectification of the exceptional disturbance
in the proportion will be brought about simply by the
continual influx of fresh terms in the series. These will in
the long run neutralize the disturbance, not by any special
adaptation, as it were, for the purpose, but by the mere
weight of their overwhelming numbers. At every stage
therefore, in the succession, whatever might have been the
number and nature of the preceding terms, it will still be
true to say that one in ten of the terms will be an X.
If we had to do only with a finite number of terms,
however large that number might be, such a disturbance as
we have spoken of would, it is true, need a special alteration
in the subsequent proportions to neutralize its effects. But
when we have to do with an infinite number of terms, this
is not the case; the 'limit' of the series, which is what we
then have to deal with, is unaffected by these temporary
disturbances. In the continued progress of the series we
shall find, as a matter of fact, more and more of such disturbances,
and these of a more and more exceptional character.
But whatever the point we may occupy at any time, if we
look forward or backward into the indefinite extension of the
series, we shall still see that the ultimate limit to the proportion
in which its terms are arranged remains the same;
and it is with this limit, as above mentioned, that we are
concerned in the strict rules of Probability.
The most familiar example, perhaps, of this kind is that
of tossing up a penny. Suppose we have had four heads in
succession; people[11] have tolerably realized by now that 'head
the fifth time' is still an even chance, as 'head' was each
time before, and will be ever after. The preceding paragraph
explains how it is that these occasional disturbances
in the average become neutralized in the long run.
29. (3) There are other cases which, though rare,
are by no means unknown, in which such an inference as
that obtained from the Rule of Succession would be the direct
reverse of the truth. The oftener a thing happens, it
may be, the more unlikely it is to happen again. This is the
case whenever we are drawing things from a limited source
(as balls from a bag without replacing them), or whenever
the act of repetition itself tends to prevent the succession
(as in giving false alarms).
I am quite ready to admit that we believe the results described
in the last two classes on the strength of some such
general Inductive rule, or rather principle, as that involved
in the first. But it would be a great error to confound this
with an admission of the validity of the rule in each special
instance. We are speaking about the application of the rule
to individual cases, or classes of cases; this is quite a distinct
thing, as was pointed out in a previous chapter, from
giving the grounds on which we rest the rule itself. If a
man were to lay it down as a universal rule, that the testimony
of all persons was to be believed, and we adduced an
instance of a man having lied, it would not be considered
that he saved his rule by showing that we believed that it
was a lie on the word of other persons. But it is perfectly
consistent to give as a merely general, but not universal,
rule, that the testimony of men is credible; then to separate
off a second class of men whose word is not to be trusted,
and finally, if any one wants to know our ground for the
second rule, to rest it upon the first. If we were speaking
of _necessary_ laws, such a conflict as this would be as
hopeless as the old 'Cretan' puzzle in logic; but in instances
of Inductive and Analogical extension it is perfectly
harmless.
30. A familiar example will serve to bring out the
three different possible conclusions mentioned above. We
have observed it rain on ten successive days. A and B conclude
respectively for and against rain on the eleventh day;
C maintains that the past rain affords no data whatever for
an opinion. Which is right? We really cannot determine
_à priori_. An appeal must be made to direct observation, or
means must be found for deciding on independent grounds
to which class we are to refer the instance. If, for example,
it were known that every country produces its own rain, we
should choose the third rule, for it would be a case of drawing
from a limited supply. If again we had reasons to believe
that the rain for our country might be produced anywhere
on the globe, we should probably conclude that the past
rainfall threw no light whatever on the prospect of a continuance
of wet weather, and therefore take the second.
Or if, finally, we knew that rain came in long spells or seasons,
as in the tropics, then the occurrence of ten wet days
in succession would make us believe that we had entered on
one of these seasons, and that therefore the next day would
probably resemble the preceding ten.
Since then all these forms of such an Inductive rule are
possible, and we have often no _à priori_ grounds for preferring
one to another, it would seem to be unreasonable to attempt
to establish any universal formula of anticipation. All that
we can do is to ascertain what are the circumstances under
which one or other of these rules is, as a matter of fact,
found to be applicable, and to make use of it under those
circumstances.
31. (VII.) In the cases discussed in (V.) the almost
infinitely small chances with which we were concerned were
rightly neglected from all practical consideration, however
proper it might be, on speculative grounds, to keep our minds
open to their actual existence. But it has often occurred to
me that there is a common error in neglecting to take them into
account when they may, though individually small, make up
for their minuteness by their number. As the mathematician
would express it, they may occasionally be capable of being
_integrated_ into a finite or even considerable magnitude.
For instance, we may be confronted with a difficulty out
of which there appears to be only one appreciably possible
mode of escape. The attempt is made to force us into
accepting this, however great the odds apparently are against
it, on the ground that improbable as it may seem, it is at
any rate vastly more probable than any of the others. I
can quite admit that, on practical grounds, we may often find
it reasonable to adopt this course; for we can only _act_ on
one supposition, and we naturally and rightly choose, out
of a quantity of improbabilities, the least improbable. But
when we are not forced to act, no such decisive preference is
demanded of us. It is then perfectly reasonable to refuse
assent to the proposed explanation; even to say distinctly
that we do not believe it, and at the same time to decline,
at present, to accept any other explanation. We remain, in
fact, in a state of suspense of judgment, a state perfectly
right and reasonable so long as no action demanding a specific
choice is forced upon us. One alternative may be
decidedly probable as compared with any other individually,
but decidedly improbable as compared with all others collectively.
This in itself is intelligible enough; what people
often fail to see is that there is no necessary contradiction
between saying and feeling this, and yet being prepared
vigorously to act, when action is forced upon us, as though
this alternative were really the true one.
32. To take a specific instance, this way of regarding
the matter has often occurred to me in disputes upon 'Spiritualist'
manifestations. Assent is urged upon us because,
it is said, no other possible solution can be suggested. It
may be quite true that apparently overwhelming difficulties
may lie as against each separate alternative solution; but is
it always sufficiently realized how numerous such solutions
may be? No matter that each individually may be almost
incredible: they ought all to be massed together and thrown
into the scale against the proffered solution, when the only
question asked is, Are we to accept this solution? There is
no unfairness in such a course. We are perfectly ready to
adopt the same plan against any other individual alternative,
whenever any person takes to claiming this as _the_
solution of the difficulty. We are looking at the matter
from a purely logical point of view, and are quite willing, so
far, to place every solution, spiritualist or otherwise, upon the
same footing. The partisans of every alternative are in
somewhat the same position as the members of a deliberative
assembly, in which no one will support the motion of any
other member. Every one can aid effectively in rejecting
every other motion, but no one can succeed in passing his
own. Pressure of urgent necessity may possibly force them
out of this state of practical inaction, by, so to say, breaking
through the opposition at some point of least resistance; but
unless aided by some such pressure they are left in a state of
hopeless dead-lock.
33. Assuming that the spiritualistic solution admits
of, and is to receive, scientific treatment, this, it seems to
me, is the conclusion to which one might sometimes be led
in the face of the evidence offered. We might have to say to
every individual explanation, It is incredible, I cannot accept
it; and unless circumstances should (which it is hardly possible
that they should) force us to a hasty decision,--a decision,
remember, which need indicate no preference of the
judgment beyond what is just sufficient to turn the scale in its
favour as against any other single alternative,--we leave the
matter thus in abeyance. It will very likely be urged that
one of the explanations (assuming that all the possible ones
had been included) must be true; this we readily admit.
It will probably also be urged that (on the often-quoted
principle of Butler) we ought forthwith to accept the one
which, as compared with the others, is the most plausible,
whatever its absolute worth may be. This seems distinctly
an error. To say that such and such an explanation is the
one we should accept, _if_ circumstances compelled us to anticipate
our decision, is quite compatible with its present
rejection. The only rational position surely is that of
admitting that the truth is somewhere amongst the various
alternatives, but confessing plainly that we have no such
preference for one over another as to permit our saying anything
else than that we disbelieve each one of them.
34. (VIII.) The very common fallacy of 'judging by
the event,' as it is generally termed, deserves passing notice
here, as it clearly belongs to Probability rather than to Logic;
though its nature is so obvious to those who have grasped the
general principles of our science, that a very few words of
remark will suffice. In one sense every proposition must
consent to be judged by the event, since this is merely, in
other words, submitting it to the test of experience. But
there is the widest difference between the test appropriate
to a universal proposition and that appropriate to a merely
proportional or statistical one. The former is subverted by a
single exception; the latter not merely admits exceptions,
but implies them. Nothing, however, is more common than
to blame advice (in others) because it has happened to turn
out unfortunately, or to claim credit for it (in oneself) because
it has happened to succeed. Of course if the conclusion was
avowedly one of a probable kind we must be prepared with
complacency to accept a hostile event, or even a succession of
them; it is not until the succession shows a disposition to
continue over long that suspicion and doubt should arise, and
then only by a comparison of the degree of the assigned
probability, and the magnitude of the departure from it
which experience exhibits. For any single failure the reply
must be, 'the advice was sound' (supposing, that is, that it
was to be justified in the long run), 'and I shall offer it again
under the same circumstances.'
35. The distinction drawn in the above instance
deserves careful consideration; for owing to the wide difference
between the kind of propositions dealt with in Probability
and in ordinary Logic, and the consequent difference in
the nature of the proof offered, it is quite possible for arguments
of the same general appearance to be valid in the
former and fallacious in the latter, and conversely.
For instance, take the well-known fallacy which consists
in simply converting a universal affirmative, i.e. in passing
from All A is B to All B is A. When, as in common Logic,
the conclusion is to be as certain as the premise, there is not
a word to be said for such a step. But if we look at the
process with the more indulgent eye of Induction or Probability
we see that a very fair case may sometimes be made
out for it. The mere fact that 'Some B is A' raises a certain
presumption that any particular B taken at random will
be an A. There is some reason, at any rate, for the belief,
though in the absence of statistics as to the relative frequency
of A and B we are unable to assign a value to this
belief. I suspect that there may be many cases in which a
man has inferred that some particular B is an A on the
ground that All A is B, who might justly plead in his behalf
that he never meant it to be a necessary, but only a probable
inference. The same remarks will of course apply
also to the logical fallacy of Undistributed Middle.
Now for a case of the opposite kind, i.e. one in which
Probability fails us, whereas the circumstances seem closely
analogous to those in which ordinary inference would be
able to make a stand. Suppose that I know that one letter
in a million is lost when in charge of the post. I write to a
friend and get no answer. Have I any reason to suppose
that the fault lies with him? Here is an event (viz. the
loss of the letter) which has certainly happened; and we suppose
that, of the only two causes to which it can be assigned,
the 'value,' i.e. statistical frequency, of one is accurately
assigned, does it not seem natural to suppose that something
can be inferred as to the likelihood that the other cause had
been operative? To say that nothing can be known about
its adequacy under these circumstances looks at first sight
like asserting that an equation in which there is only one
unknown term is theoretically insoluble.
As examples of this kind have been amply discussed in
the chapter upon Inverse rules of Probability I need do no
more here than remind the reader that no conclusion whatever
can be drawn as to the likelihood that the fault lay
with my friend rather than with the Post Office. Unless we
either know, or make some assumption about, the frequency
with which he neglects to answer the letters he receives, the
problem remains insoluble.
The reason why the apparent analogy, indicated above,
to an equation with only one unknown quantity, fails to hold
good, is that for the purposes of Probability there are really
_two_ unknown quantities. What we deal with are proportional
or statistical propositions. Now we are only told that
in the instance in question the letter was lost, not that they
were found to be lost in such and such a proportion of cases.
Had this latter information been given to us we should really
have had but one unknown quantity to determine, viz. the
relative frequency with which my correspondent neglects to
answer his letters, and we could then have determined this
with the greatest ease.
1. Discussed by Mr F. Y. Edgeworth, in the _Phil. Mag._ for April,
1887.
2. _Journal of the Statistical Soc._ (Vol. XLII. p. 328) Dare one
suspect a joke?
3. It appears to have been long known to gamblers under the name of
the _Martingale_. There is a paper by Babbage (_Trans. of Royal
Soc. of Edinburgh_, for 1823) which discusses certain points
connected with it, but scarcely touches on the subject of the
sections which follow.
4. Attention will be further directed to this distinction in the
chapter on Insurance and Gambling.
5. As by Prévost in the _Bibliothèque Universelle de Genève_,
Oct. 1829. The explanation is noted, and apparently accepted, by
Quetelet (_Physique Sociale_, I. 171).
6. _Essay on Probabilities_, p. 126.
7. This theoretical or absolute neglect of what is very rare must not
be confused with the practical neglect sometimes recommended by
astronomical and other observers. A criterion, known as Chauvenet's,
for indicating the limits of such rejection will be found described
in Mr Merriman's _Least Squares_ (p. 166). But this rests on the
understanding that a smaller balance of error would thus result in
the long run. The very rare event is deliberately rejected, not
overlooked.
8. The process of calculation may be readily indicated. There are,
say, about 350,000 letters in the work in question. Since any of the
26 letters of the alphabet may be drawn each time, the possible
number of combinations would be 26^{350,000}; a number which, as may
easily be inferred from a table of logarithms, would demand for its
expression nearly 500,000 figures. Only one of these combinations is
favourable, if we reject variations of spelling. Hence unity divided
by this number would represent the chance of getting the desired
result by successive random selection of the required number of
350,000 letters.
If this chance is thought too small, and any one asks how often the
above random selection must be repeated in order to give him odds of
2 to 1 in favour of success, this also can be easily shown. If the
chance of an event on each occasion is 1/n, the chance of getting it
once at least in n trials is 1 - ((n - 1)/n)^{n}; for we shall do
this unless we fail n times running. When (as in the case in
question) n is very large, this may be shown algebraically to be
equivalent to odds of about 2 to 1. That is, when we have drawn the
requisite quantity of letters a number of times equal to the
inconceivably great number above represented, it is still only 2 to
1 that we shall have secured what we want:--and then we have to
recognize it.
9. The longest life which could reasonably be attributed to any
language would of course dwindle into utter insignificance in the
face of such periods of time as are being here arithmetically
contemplated.
10. We are here assuming of course that the ultimate limit to which
our average tends is known, either from knowledge of the causes or
from previous extensive experience. We are assuming that e.g. the
die is known to be a fair one; if this is not known but a possible
bias has to be inferred from its observed performances, the case
falls under the former head.
11. Except indeed the gamblers. According to a gambling acquaintance
whom Houdin, the conjurer, describes himself as having met at Spa,
"the oftener a particular combination has occurred the more certain
it is that it will not be repeated at the next _coup_: this is the
groundwork of all theories of probabilities and is termed the
maturity of chances" (_Card-sharping exposed_, p. 85).