is almost impossible, for instance, that any whist player should have played in any two games where the distribution of the cards was exactly the same, by pure accident (p. 217). Such a thing as a person always losing at a game of pure chance, is wholly unknown. Coincidences of this kind are not impossible, as I have said, but they are so unlikely that the lifetime of any person, or indeed the whole duration of history does not give any appreciable probability of their being encountered. Whenever we make any extensive series of trials of chance results, as in throwing a die or coin, the probability is great that the results will agree nearly with the predictions yielded by theory. Precise agreement must not be expected, for that, as the theory could show, is highly improbable. Several attempts have been made to test, in this way, the accordance of theory and experience. The celebrated naturalist, Buffon, caused the first trial to be made by a young child who threw a coin many times in succession, and he obtained 1992 tails to 2048 heads. A pupil of Professor De Morgan repeated the trial for his own satisfaction, and obtained 2044 tails to 2048 heads. In both cases the coincidence with theory is as close as could be expected, and the details may be found in De Morgan's 'Formal Logic,' p. 185.

Quetelet also tested the theory in a rather more complete manner, by placing 20 black and 20 white balls in an urn and drawing a ball out time after time in an indifferent manner, each ball being replaced before a new drawing was made. He found, as might be expected, that the greater the number of drawings made the more nearly were the white and black balls equal in number. At the termination of the experiment he had registered 2066 white and 2030 black balls, the ratio being 1'02'.

i 'Letters on the Theory of Probabilities,' translated by Downes, 1849, pp. 36, 37.

I have made a series of experiments in a third manner, which seemed to me even more interesting, and capable of more extensive trial. Taking a handful of ten coins, usually shillings, I threw them up time after time, and registered the numbers of heads which appeared each time. Now the probability of obtaining 10, 9, 8, 7, &c., heads is proportional to the number of combinations of 10, 9, 8, 7, &c., things out of 10 things. Consequently the results ought to approximate to the numbers in the eleventh line of the Arithmetical Triangle. I made altogether 2048 throws, in two sets of 1024 throws each, and the numbers obtained are given in the following

The whole number of single throws of coins amounted to 10x 2048 or 20,480 in all, one half of which or 10,240 should theoretically give head. The total number of heads obtained was actually 10,353, or 5222 in the first series, and 5131 in the second. The coincidence with theory is pretty close, but considering the large number of throws there is some reason to suspect a tendency in favour of heads.

The special interest of this trial consists in the exhibition, in a practical form, of the results of Bernouilli's theorem, and the law of error or divergence from the

mean to be afterwards more fully considered. It illustrates the connection between combinations and permutations, which is exhibited in the Arithmetical Triangle, and which underlies many of the most important theorems of science.

Probable Deductive Arguments.

With the aid of the theory of probabilities, we may extend the sphere of deductive argument. Hitherto we have treated propositions as certain, and on the hypothesis of certainty have deduced conclusions equally certain. But the information on which we reason in ordinary life is seldom or never certain, and almost all reasoning is really a question of probability. We ought therefore to be fully aware of the mode and degree in which the forms of deductive reasoning are affected by the theory of probability, and many persons might be surprised at the results which must be admitted. Many controversial writers appear to consider, as De Morgan remarked, that an inference from several equally probable premises is itself as probable as any of them, but the true result is very different. If a fact or argument involves many propositions, and each of them is uncertain, the conclusion will be of very little force.

The truth of a conclusion may be regarded as a compound event, depending upon the premises happening to be true; thus, to obtain the probability of the conclusion, we must multiply together the fractions expressing the probabilities of the premises. Thus, if the probability is that A is B, and also that B is C, the conclusion that A is C, on the ground of these premises, is or 1. Similarly if there be any number of premises requisite to

kEncyclopædia Metrop.' art. Probabilities, p. 396.

the establishment of a conclusion and their probabilities be m, n, p, q, r, &c., the probability of the conclusion on the ground of these premises is m×nxpxq× r × . . . . . . . This product has but a small value, unless each of the quanties m, n, &c., be nearly unity.

But it is particularly to be noticed that the probability thus calculated is not the whole probability of the conclusion, but that only which it derives from the premises in question. Whately's remarks on this subject might mislead the reader into supposing that the calculation is completed by multiplying together the probabilities of the premises. But it has been fully explained by De Morgan m that we must take into account the antecedent probability of the conclusion; A may be C for other reasons besides its being B, and as he remarks, 'It is difficult, if not impossible, to produce a chain of argument of which the reasoner can rest the result on those arguments only.' We must also bear in mind that the failure of one argument does not, except under special circumstances, disprove the truth of the conclusion it is intended to uphold, otherwise there are few truths which could survive the ill considered arguments adduced in their favour. But as

a rope does not necessarily break because one strand in it is weak, so a conclusion may depend upon an endless number of considerations besides those immediately in view. Even when we have no other information we must not consider a statement as devoid of all probability. The true expression of complete doubt is a ratio of equality between the chances in favour of and against it, and this ratio is expressed in the probability.

Now if A and C are wholly unknown things, we have no reason to believe that A is C rather than A is not C. The antecedent probability is then. If we also have the

1 Elements of Logic,' Book III, sections, 11 and 18.

m Encyclopædia Metrop.' art. Probabilities, p. 400.

probabilities that A is B, and that B is C, 1, we have no right to suppose that the probability of A being C is reduced by the argument in its favour. If the conclusion is true on its own grounds, the failure of the argument does not affect it; thus its total probability is its antecedent probability, added to the probability that this failing, the new argument in question establishes it. There is a probability that we shall not require the special argument; a probability that we shall, and a probability that the argument does in that case establish it. Thus the complete result is + × 1, or §. In general language, if a be the probability formed on a particular argument, and e the antecedent probability, then the general result is

I-(1-a)(1c), or a+c-ac.

We may put it still more generally in this way :-Let a, b, c, d, &c., be the probabilities of a conclusion grounded on various arguments or considerations of any kind. It is only when all the arguments fail that our conclusion proves finally untrue; the probabilities of each failing are respectively 1-a, 1−b, 1−c, &c.; the probability that they will all fail (1− a)(1 − b)(1 −c)...; therefore the probability that the conclusion will not fail is 1 − (1 − a)(1 − b)(1c)...&c. On this principle it follows that every argument in favour of a fact, however flimsy and slight, adds probability to it. When it is unknown whether an overdue vessel has foundered or not, every slight indication of a lost vessel will add some probability to the belief of its loss, and the disproof of any particular evidence will not disprove the event.

We must apply these principles of evidence with great care, and observe that in a great proportion of cases the adducing of a weak argument does tend to the disproof of its conclusion. The assertion may have in itself great inherent improbability as being opposed to other evidence

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