principle, of which the following may be taken as an equivalent.
If we distribute the favourable cases into several groups the probability
of the event will be sum of the probabilities pertaining to
each group.^{[1]}

10. Another important instance of unverified probabilities occurs
when it is assumed without specific experience that one phenomenon
is independent of another in such wise that the probability of a double
event is equal to the product of the one event multiplied by the probability
of the other—as in the instance already given of two aces
occurring. The assumption has been verified with respect to “runs”
in some games of chance;^{[2]} but it is legitimately applied far beyond
those instances. The proposition that very long runs of particular
digits, *e.g.* of 7, may be expected in the development of a constant
like π—*e.g.* a run of six consecutive *sevens* if the expansion of the
constant was carried to a million places of decimals—may be given
as an instance in which our conviction greatly transcends specific
verification. In the calculation of probable, and improbable, errors,
it^{[3]} has to be assumed without specific verification that the observations
on which the calculation is based are independent of each other
in the sense now under consideration. With these explanations we
may accept Laplace's third principle “If the events are independent
of each other the probability of their concurrence (*l'existence de leur*
*ensemble*) is the product of their separate probabilities.”^{[4]}

11. *Interdependent Probabilities*.—Among the principles of probabilities
it is usual to enunciate, after Laplace, several other
propositions.^{[5]} But these may here be rapidly passed over as they do
not seem to involve any additional philosophical difficulty.

12. It has been shown that when two events are independent
of each other the product of their separate probabilities forms the
probability of their concurrence. It follows that the probability
of the double event divided by the probability of either, say the first,
component gives the probability of the other, the second component
event. The quotient, we might say, is the probability that when
the first event has occurred, the second will occur. The proposition
in this form is true also of events which are not independent of one
another. Laplace exemplifies the composition of such interdependent
probabilities by the instance of three urns, A, B, C, about which it
is known that two contain only white balls and one only black balls.^{[6]}
The probability of drawing a white ball from an assigned urn, say C,
is ⅔. The probability that, a white ball having been drawn from C,
a ball drawn from B will be white, is ½. Therefore the probability
of the double event drawing a white ball from C and also from B is
⅔ × ½ or ⅓. The question now arises. Supposing we know only
the probability of the double event, which probability we will call
BC, and the probability of one of them, say [C] (but not, as in the
case instanced, the mechanism of their interdependence); what can
we infer about the probability [B] of the other event (an event such
as in the above instance drawing a white ball from the urn B)—the
separate probability irrespective of what has happened as to the urn
C? We cannot in general say that [B] = [BC] divided by [C] but
rather that quotient × *k*, where *k* is an unknown coefficient which
may be either positive or negative. It might, however, be improper
to treat *k* as zero on the ground that it is equally likely (in the long
run of similar data) to be positive or negative. For given values
of [BC] and [C], *k* has not this equiprobable character, since its
positive and negative ranges are not in general equal; as appears
from considering that [B] cannot be less than [BC], nor greater
than unity.^{[7]}

13. *Probability of Causes and Future Effects*.—The first principles
which have been established afford an adequate ground for the
reasoning which is described as deducing the probability of a cause
from an observed event.^{[8]} If with the poet^{[9]} we may represent a
perfect mixture by the waters of the Po in which the “two Doras”
and other tributaries are indiscriminately commingled, there is no
great difference in respect of definition and deduction between the
probability that a certain particle of water should have emanated
from a particular source, or should be discharged through a particular
mouth of the river. “This principle,” we may say with De Morgan,
“of the retrospective or ‘inverse’ probability is not essentially
different from the one first stated (Principle I.).”^{[10]} Nor is a new
first principle necessarily involved when after ascending from an
effect to a cause we descend to a collateral effect.^{[11]} It is true that in
the investigation of causes it is often necessary to have recourse
to the unverified species of probability. An instance has already
been given of several approximately equiprobable causes, the several
values of a quantity under measurement, from one of which the
observed phenomena, the given set of observations, must have, so
to speak, emanated. A simpler instance of two alternative causes
occurs in the investigation which J. S. Mill^{[12]} has illustrated—whether
an event, such as a succession of aces, has been produced by a particular
cause, such as loading of the die, or by that mass of “fleeting
causes” called chance. It is sufficient for the argument that the
“a priori” probabilities of the alternatives should not be very
unequal.^{[13]}

14. (2) *Whether Credibility is Measurable*.—The domain of probabilities
according to some authorities does not extend much, if at all,
beyond the objective phenomena which have been described in the
preceding paragraphs. The claims of the science to measure the
subjective quantity, degree of belief, are disallowed or minimized.
Belief, it is objected, depends upon a complex of perceptions and
emotions not amenable^{[14]} to calculus. Moreover, belief is not credibility;
even if we do believe with more or less confidence in exact
conformity with the measure of probability afforded by the calculus,
*ought* we so to believe? In reply it must be admitted that many of
the beliefs on which we have to act are not of the kind for which
the calculus prescribes. It was absurd of Craig^{[15]} to attempt to evaluate
the credibility of the Christian religion by mathematical calculation.
But there seem to be a number of simpler cases of which we
may say with De Morgan^{[16]} “that in the universal opinion of those
who examine the subject, the state of mind to which a person *ought*
to be able to bring himself” is in accordance with the regulation
measure of probability. If in the ordeal to which Portia's suitors
were subjected there had been a picture of her not in one only, but
in two of the caskets, then—though the judgment of the principal
parties might be distorted by emotion—the impartial spectator
would normally expect with greater confidence than before that at
any particular trial a casket containing the likeness of the lady
would be chosen. So the indications of a thermometer may not
correspond to the sensations of a fevered patient, but they serve to
regulate the temperature of a public library so as to secure the comfort
of the majority. This view does not commit us to the quantitative
precision of De Morgan that in a case such as above supposed
we ought to “look three times as confidently upon the arrival as
upon the non-arrival” of the event.^{[17]} Two or three roughly distinguished
degrees of credibility—very probable, as probable as not,
very improbable, practically impossible—suffice for the more
important applications of the calculus. Such is the character of
the judgments which the calculus enables us to form with respect to
the occurrence of a certain difference between the real value of any
quantity under measurement and the value assigned to it by the
measurement. The confidence that the constants which we have
determined are accurate within certain limits is a subjective feeling
which cannot be dislodged from an important part of probabilities.^{[18]}
This sphere of subjective probability is widened by the latest developments
of the science^{[19]} so far as they add to the number of constants
for which it is important to determine the probable—and improbable—error.
For instance, a measure of the deviation of observations
from an average or mean value was required by the older writers
only as subordinate to the determination of the mean, but now this
“standard deviation” (below, par. 98) is often treated as an entity
for which it is important to discover the limits of error.^{[20]} Some of
the newer methods may also serve to countenance the measurement
of subjective quantity, in so far as they successfully apply the
calculus to quantities not admitting of a precise unit, such as colour

- ↑ Bertrand on “Probabilités composées,”
*op. cit.*art. 23. - ↑ In some of the experiences referred to at par. 5.
- ↑ See below pars. 132, 159.
- ↑
*Op. cit.*Introduction. - ↑ There is a good statement of them in Boole's
*Laws of Thought*, ch. xvi. § 7. Cf. De Morgan “Theory of Probabilities” (*Encyc.**Metrop.*), §§ 12 seq. - ↑ Laplace,
*op. cit.*Introduction,*IV*; cf.^{e}Principe*V*and liv., II. ch. i. § 1.^{e}Principe - ↑ In such a case there seems to be a propriety in expressing the
indeterminate element in our data, not as above, but as proposed
by Boole in his remarkable
*Laws of Thought*, ch. xvii., ch. xviii., § 1 (cf.*Trans. Edin. Roy. Soc.*, (1857), vol. xxi.; and*Trans. Roy. Soc.*, 1862, vol. ix., vol. clii. pt. i. p. 251); the undetermined constant now representing the probability that if the event C does not occur the event B will. The values of*this*constant—in the absence of specific data, and where independence is not presumable—are, it should seem, equally distributed between the values 0 and 1. Cf. as to Boole's Calculus,*Mind*,*loc. cit.*, ix. 230 seq. - ↑ Laplace's
*Sixth*Principle. - ↑ Manzoni.
- ↑ De Morgan,
*Theory of Probabilities*, § 19; cf. Venn,*Logic of**Chance*, ch. vii. § 9; Edgeworth, “On the Probable Errors of Frequency Constants,”*Journ. Stat. Soc.*(1908), p. 653. The essential symmetry of the inverse and the direct methods is shown by an elegant proof which Professor Cook Wilson has given for the received rules of inverse probability (*Nature*, 1900, Dec. 13). - ↑ Laplace's
*Seventh*Principle. - ↑
*Logic*, book III., ch. xviii. § 6. - ↑ Cf. above, par. 8; below, par. 46.
- ↑ Cf. Venn,
*Logic of Chance*, p. 126. - ↑ See the reference to Craig in Todhunter,
*History . . . of Probability*. - ↑
*Formal Logic*, p. 173. - ↑ Ibid. Cf. “Theory of Probabilities” (
*Encyc. Metrop.*), note to § 5, “Wherever the term greater or less can be applied there twice, thrice, &c., can be conceived, though not perhaps measured by us.” - ↑ It is well remarked by Professor Irving Fisher (
*Capital and**Income*, 1907, ch. xvi.), that Bernoulli's theorem involves a “subjective” element a “psychological magnitude.” The remark is applicable to the general theory of error of which the theorem of Bernoulli is a particular case (see below, pars. 103, 104). - ↑ In the hands of Professor Karl Pearson, Mr Sheppard and Mr Yule. Cf. par. 149, below.
- ↑ Cf. Edgeworth,
*Journ. Stat. Soc.*(Dec. 1908).