Encyclopaedia Britannica > Volume 19, PHY-PRO

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PROBABILITY
THE mathematical theory of probability is a science
which aims at reducing to calculation, where possible,
the amount of credence due to propositions or statements,
or to the occurrence of events, future or past, more especi¬
ally as contingent or dependent upon other propositions
or events the probability of which is known.
Any statement or (supposed) fact commands a certain
amount of credence, varying from zero, which means con¬
viction of its falsity, to absolute certainty, denoted by
unity. An even chance, or the probability of an event
which is as likely as not to happen, is represented by the
fraction It is to be observed that | will be the
probability of an event about which we have no knowledge
whatever, because if we can see that it is more likely to
happen than not, or less likely than not, we must be in
possession of some information respecting it. It has been
proposed to form a sort of thermometrical scale, to which
to refer the strength of the conviction we have in any
given case. Thus if the twenty-six letters of the alphabet
have been shaken together in a bag, and one letter be
drawn, we feel a very feeble expectation that A has been
the one taken. If two letters be drawn, we have still
very little confidence that A is one of them; if three be
drawn, it is somewhat stronger; and so on, till at last, if
twenty-six be drawn, we are certain of the event, that is,
of A having been taken.
Probability, which necessarily implies uncertainty, is a
consequence of our ignorance. To an omniscient Being
there can be none. Why, for instance, if we throw up a
shilling, are we uncertain whether it will turn up head or
tail ? Because the shilling passes, in the interval, through
a series of states which our knowledge is unable to predict
or to follow. If we knew the exact position and state of
motion of the coin as it leaves our hand, the exact value
of the final impulse it receives, the laws of its motion as
affected by the resistance of the air and gravity, and
finally the nature of the ground at the exact spot where it
falls, and the laws regulating the collision between the
two substances, we could predict as certainly the result
of the toss as we can which letter of the alphabet will be
drawn after twenty-five have been taken and examined.
The probability, or amount of conviction accorded to
any fact or statement, is thus essentially subjective, and
varies with the degree of knowledge of the mind to which
the fact is presented (it is often indeed also influenced by
passion and prejudice, which act powerfully in warping
the judgment),—so that, as Laplace observes, it is affected
partly by our ignorance partly by our knowledge. Thus,
if the question were put, Is lead heavier than silver?
some persons would think it is, but would not be surprised
if they were wrong; others would say it is lighter; while
to a worker in metals probability would be superseded by
certainty. Again, to take Laplace’s illustration, there are
three urns A, B, C, one of which contains black balls, the
other two white balls ; a ball is drawn from the urn C, and
we want to know the probability that it shall be black.
If we do not know which of the urns contains the black
balls, there is only one favourable chance out of three, and
the probability is said to be J. But if a person knows
that the urn A contains white balls, to him the uncertainty
is confined to the urns B and C, and therefore the proba¬
bility of the same event is Finally to one who had
found that. A and B both contained white balls, the
probability is converted into certainty.
In common language, an event is usually said to be
likely or probable if it is more likely to happen than not,
or when, in mathematical language, its probability exceeds
It ; and it is said to be improbable or unlikely when its
probability is less than Not that this sense is always
adhered to; for, in such a phrase as “ It is likely to
thunder to-day,” we do not mean that is more likely than
not, but that in our opinion the chance of thunder is
greater than usual; again, “ Such a horse is likely to win
the Derby,” simply means that he has the best chance,
though according to the betting that chance may be only
Such unsteady and elliptical employment of words
has of course to be abandoned and replaced by strict
definition, at least mentally, when they are made the
subjects of mathematical analysis. Certainty, or absolute
conviction, also, as generally understood, is different from
the mathematical sense of the word certainty. It is very
difficult and often impossible, as is pointed out in the
celebrated Grammar of Assent, to draw out the grounds
on which the human mind in each case yields that con¬
viction, or assent, which, according to Newman, admits of
no degrees, and either is entire or is not at all.1 If, when
walking on the beach, we find the letters “ Constantinople ”
traced on the sand, we should feel, not a strong impression,
but absolute certainty, that they were characters not
drawn at random, but by one acquainted with the word
so spelt. Again, we are certain of our own death as a
future event; we are certain, too, that Great Britain is an
island; yet in all such cases it would be very difficult,
even for a practised intellect, to present in logical form
the evidence, which nevertheless has compelled the mind
in each instance to concede the point.2 Mathematical
certainty, which means that the contrary proposition is
inconceivable, is thus different, though not perhaps as
regards the force of the mental conviction, from moral or
practical certainty. It is questionable whether the former
kind of certainty is not entirely hypothetical, and whether
it is ever attainable in any of the affairs or events of the
real world around us. The truth of no conclusion can rise
above that of the premises, of no theorem above that of
the data. That two and two make four is an incon¬
trovertible truth ; but before applying even it to a concrete
instance we have to be assured that there were really
two in each constituent group; and we can hardly have
mathematical certainty of this, as the strange freaks of
memory, the tricks of conjurors, &c., have often made
apparent.
There is no more remarkable feature in the mathematical
theory of probability than the manner in which it has
been found to harmonize with, and justify, the conclusions
to which mankind have been led, not by reasoning, but by
instinct and experience, both of the individual and of the
race. i At the same time it has corrected, extended, and
invested them with a definiteness and precision of which
these crude, though sound, appreciations of common sense
were till then devoid. Even in cases where the theoretical
result appears to differ from the common-sense view, it
often happens that the latter may, though perhaps
unknown to the mind itself, have taken account of
circumstances in the case omitted in the data of the
1 “ There is a sort of leap which most men make from a high pro¬
bability to absolute assurance . . . analogous to the sudden consilience,
or springing into one, of the two images seen by binocular vision, when
gradually brought within a certain proximity.”—Sir J. Herschel, in
Edin. Review, July 1850.
2 Archbishop Whately’s jeu d'esprit, Historic Doubts respecting
Napoleon Bonaparte, is a good illustration of the difficulties there may
be in proving a conclusion the certainty of which is absolute.