would lead us to a probability of 1 or of any greater number, results which could have no meaning whatever. The probability we wish to calculate is that of one head in two throws, but in our addition we have included the case in which two heads appear. The true result is + 1 × 1 or, or the probability of head at the first throw, added to the exclusive probability that if it does not come at the first, it will come at the second. The greatest difficulties of the theory arise from the confusion of exclusive and unexclusive alternatives. I may remind the reader that the possibility of unexclusive alternatives was a point previously discussed (p. 68), and to the reasons then given for considering alternation as logically unexclusive, may be added the existence of these difficulties in the theory of probability. The erroneous result explained above really arose from overlooking the fact that the expression "head first throw or head second throw" might include the case of head at both throws.

The Logical Alphabet in questions of Probability.

When the probabilities of certain simple events are given, and it is required to deduce the probabilities of compound events, the Logical Alphabet may give assistance, provided that there are no special logical conditions so that all the combinations are possible. Thus, if there be three events, A, B, C, of which the probabilities are, a, ß, y, then the negatives of those events, expressing the absence. of the events, will have the probabilities I-a, I - ẞ, I — y. We have only to insert these values for the letters of the combinations and multiply, and we obtain the probability of each combination. Thus the probability of ABC is apy; of Abc, a(I – B)(1 − y).

We can now clearly distinguish between the probabilities of exclusive and unexclusive events. Thus, if A and B. are events which may happen together like rain and high. tide, or an earthquake and a storm, the probability of A or B happening is not the sum of their separate probabilities. For by the Laws of Thought we develop AB into AB | Ab | aB, and substituting a and 6, the probabilities of A and B respectively, we obtain a.ẞ + a.(1 − ẞ) + (1a).ẞ or a + B-a.ß. But if events are incompossible

or incapable of happening together, like a clear sky and rain, or a new moon and a full moon, then the events are not really A or B, but A not-B, or B not-A, or in symbols. Ab | aB. Now if we take μ probability of Ab and probability of aB, then we may add simply, and the probability of Ab | aB is μ + v.

V =

=

Let the reader carefully observe that if the combination AB cannot exist, the probability of Ab is not the product of the probabilities of A and b. When certain combinations are logically impossible, it is no longer allowable to substitute the probability of each term for the term, because the multiplication of probabilities presupposes the independence of the events. A large part of Boole's Laws of Thought is devoted to an attempt to overcome this difficulty and to produce a General Method in Probabilities by which from certain logical conditions and certain given probabilities it would be possible to deduce the probability of any other combinations of events under those conditions. Boole pursued his task with wonderful ingenuity and power, but after spending much study on his work, I am compelled to adopt the conclusion that his method is fundamentally erroneous. As pointed out by Mr. Wilbraham,1 Boole obtained his results by an arbitrary assumption, which is only the most probable, and not the only possible assumption. answer obtained is therefore not the real probability, which is usually indeterminate, but only, as it were, the most probable probability. Certain problems solved by Boole are free from logical conditions and therefore may admit of valid answers. These, as I have shown,2 may be solved by the combinations of the Logical Alphabet, but the rest of the problems do not admit of a determinate answer, at least by Boole's method.

Comparison of the Theory with Experience.

The

The Laws of Probability rest upon the fundamental principles of reasoning, and cannot be really negatived by any

1 Philosophical Magazine, 4th Series, vol. vii. p. 465; vol. viii. P. 91.

Memoirs of the Manchester Literary and Philosophical Society, 3rd Series, vol. iv. p. 347.

possible experience. It might happen that a person should always throw a coin head uppermost, and appear incapable of getting tail by chance. The theory would not be falsified, because it contemplates the possibility of the most extreme runs of luck. Our actual experience might be counter to all that is probable; the whole course of events might seem to be in complete contradiction to what we should expect, and yet a casual conjunction of events might be the real explanation. It is just possible that some regular coincidences, which we attribute to fixed laws of nature, are due to the accidental conjunction of phenomena in the cases to which our attention is directed. All that we can learn from finite experience is capable, according to the theory of probabilities, of misleading us, and it is only infinite experience that could assure us of any inductive truths.

At the same time, the probability that any extreme runs of luck will occur is so excessively slight, that it would be absurd seriously to expect their occurrence. It 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. 191). 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 shows, is highly improbable. Several attempts have been made to test, in this way, the accordance of theory and experience. 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 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 102.1

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 table::

The whole number of single throws to 10 x 2048, or 20,480 in all, one 10,240 should theoretically give head.

of coins amounted half of which or The total number

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

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 Bernoulli'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 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 deductive reasoning is affected by the theory of probability, and many persons may be surprised at the results which must be admitted. Some controversial writers appear to consider, as De Morgan remarked,1 that an inference from several equally probable premises is itself as probable as any of them, but the true result is very different. If an argument involves many propositions, and each of them is uncertain, the conclusion will be of very little force.

The validity 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. 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, isor. Similarly if there be any number of premises requisite to the establish

1

Encyclopædia Metropolitana, art. Probabilities, p. 396.

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