Previously to laying down a few general rules, we shall notice the error to which persons who have not thought on the subject are most : this is a confusion between the sort of confidence which is to ___„/../ be given to a result of the theory of probabilities, and that which is claimed by actual demonstration. 3Iany persons are not aware that out of mathematics the greater number of conclusions are probable results : many of them, it is true, so highly probable that their chance of falsehood does not amount to that of drawing a black ball from among a million of white ones; but still not absolutely demonstrated. These highly probable results (so probable that the word probable in its common sense is weak as applied to them) form the ordinary know ledge of common life, and being practical certainties, are considered and mentioned as certainties, the imperceptibly small chance that they may not be true being disregarded. Hence it happens that when a result of this theory is announced, with its proper chance annexed, and though the probability of its truth is so high that it may rank with the moral certainties of ordinary life, there is a morbid disposition to dwell rather upon the one way in which the proposition may fail, than upon the million of equally possible ways in which it may be true. Thus if it be said that it is ten millions to one that r will happen, and not Q, therefore it is morally certain that P will happen— it is objected, But how de we know that the very next event will not be precisely the one of ten millions which is q and not r / The answer is, we do not know it in the absolute sense of the word; if we thus knew it, it would be a certainty that P would happen, and whether P would happen or nut would not be a question for the theory of proba bilities at all : but we do know it. in the common sense of the word, since there are hundreds of conclusions which all men call knowledge, which aro not so probable that they can be reasonably shown to have ten millions to ono in their favour.
Another way in which the confusion we have mentioned shows itself is, in the habit of reasoning against the probable truths just alluded to, by arguments which could only be valid against an assertion that these truths were absolutely demonstrated. In compliance with the forms of language, those who advance such truths treat them as (moral) certainties : the opponent overthrows their (mathematical) certainty, and the fallacy lids in his supposing that he has thereby shown them not to possess that sort of truth which was claimed for them. For example, a medical man gives his opinion that a crime committed without any apparent motive is an indication of insanity ; a newspaper ridicules this opinion, and oaks, Are there no motives then which cannot be discovered I Now, if by epparcra was meant apparent on the surface, or with slight examination, and if by indication was both arguments very powerful, it is to be assumed that the mind does not see the ratio of the email quantities, and the question is evenly balanced.
2. No evidence ; only argument for. The odds are 1 to 1-A; taken by all the logicians who have considered the subject to be A to I -A. The first result is always greater than the second ; which means that even if the argument be invalid, the conclusion may still be true, and has some probability left.
3. A witness of given credibility does not add so much to the force of the conclusion as an argument of the same probability of validity. We introduce this assertion as one of common sense : if the witness be wrong the assertion is false : if the argument be invalid, the assertion may still be true ; consequently the argument is better than the witness. How do the formulae verify this ? As follows :-if a be the probability of both, the odds for the truth are altered by the witness in the proportion of a to 1 -a and by the argument in the greater proportion of 1 to 1-a.
4. Any argument, however slight its force, addisomething to the force of its conclusion : for alteration in the ratio of 1 to 1-a is increase, let a be ever so small. But it must be remembered that this is only true of the argument per se, and has no reference to the possi bility of the weakness of the argument being itself an argument on the other aide. If a weak argument be brought forward by a person who most probably would know the stronger ones, if there were any, the argument on the other side just spoken of may arise. On this, how ever, and other points, our space obliges us to content ourselves with the reference already made.
The various problems of which the solutions have been given are mathematical consequences of the definition of probability. Every such problem is simply one of combinations, however much the length of detail, and the number of mathematical abbreviations of the process of combining, may tend to make us lose eight of the first principle. At the same time it is found requisite to establish a few simple funda mental propositions, which we shall cite, with some consequences. As we are not writing an elementary treatise, we shall not demonstrate these propositions, referring the student to any of the modern works hereinafter cited. The probabilities of the events A, B, C, &c., are denoted by b, c, &e.
1. By a, the probability of A, is meant the fraction which the number of cases favourable to the happening of A is of the whole number of cases, that is, both of those in which A can happen, and those in which it cannot : all cases being equally likely. And the probability that A will not happen is 1 -a.
2. When A and n arc events independent of each other, so that the happening of either in no way promotes nor retards the happening of the other, the probability that both shall happen is a b ; that neither shall happen, 1 -a-b + a b; that one only shall happen, a + 6-2 a b; that one or both shall happen, a+ b -a b.
3. When A and n are mutually exclusive, that is, when if one happen the other cannot, the probability that one or other shall happen is a+ b; and that neither shall happen, 1-a-b.