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distinguished into two or more simpler events. Thus the firing of a gun may be distinguished into pulling the trigger, the fall of the hammer, the explosion of the cap, &c. In this example the simple events are not independent, because if the trigger is pulled, the other events will under proper conditions necessarily follow, and their probabilities are therefore the same as that of the first event. Events are independent when the happening of one does not render the other either more or less probable than before. Thus the death of a person is neither more nor less probable because the planet Mars happens to be visible. When the component events are independent, a simple rule can be given for calculating the probability of the compound event, thus-Multiply together the fractions expressing the probabilities of the independent component events.

The probability of throwing tail twice with a penny is, or; the probability of throwing it three times. running is, or; a result agreeing with that obtained in an apparently different manner (p. 230). In fact when we multiply together the denominators, we get the whole number of ways of happening of the compound event, and when we multiply the numerators, we get the number of ways favourable to the required event.

Probabilities may be added to or subtracted from each other under the important condition that the events in question are exclusive of each other, so that not more than one of them can happen. It might be argued that as the probability of throwing head at the first trial is, and at the second trial also, the probability of throwing it in the first two throws is +, or certainty. Not only is this result evidently absurd, but a repetition of the process 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 involved the case in which two heads also appear. The true result is + × 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. Some of the greatest difficulties of the theory and the subtlest errors 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. 81), 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 expression

Head first throw or head second throw ought to be interpreted in our logical system as including both cases at once, and so it is in practice.

Employment of the Logical Abecedarium in questions of Probability.

When the probabilities of certain events are given, and it is required to deduce the probabilities of compound events, the Logical Abecedarium may give assistance, provided that there are no special logical conditions and all the combinations are possible. Thus, if there be three events A, B, C, of which the probabilities are a, B, y, then the negatives of those events, expressing the absence of the events, will have the probabilities I-a, I-ẞ, 1–7. 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 aßy; of Abc, a(1 −ẞ)(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 A + B into AB AbaB, and substituting a and B, the probabilities of A and B respectively we obtain a.ẞ+ a. (1 − ẞ) + ( 1 − a). ẞ or a + ß-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 AbaB. Now if we take

then we may add simply, and probability of Ab 4 aBμ+v. Let the reader observe that since the combination AB cannot exist, the probability of Ab is not the product of the probabilities of A and b.

But 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 the late Dr. Boole's Laws of Thought is devoted to an attempt to overcome this difficulty and 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 8, Boole obtains his results by an arbitrary assumption, which is only the most probable, and not the only possible assump

g 'Philosophical Magazine,' 4th Series, vol. vii. p. 465; vol. viii.

p. 91.

tion. The 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 may also be solved by the simple combinations of the Abecedarium, but the remainder of the problems do not admit of a determinate answer, at least by Boole's method.

Comparison of the Theory with Experience.

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

Memoirs of the Manchester Literary and Philosophical Society,'

3rd Series, vol. iv. p. 347.