CHAPTER X.

THE THEORY OF PROBABILITY.

The subject upon which we now enter must not be regarded as an isolated and curious branch of speculation. It is the necessary basis of the judgments we make in the prosecution of science, or the decisions we come to in the conduct of ordinary affairs. As Butler truly said, “Probability is the very guide of life.” Had the science of numbers been studied for no other purpose, it must have been developed for the calculation of probabilities. All our inferences concerning the future are merely probable, and a due appreciation of the degree of probability depends upon a comprehension of the principles of the subject. I am convinced that it is impossible to expound the methods of induction in a sound manner, without resting them upon the theory of probability. Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge--knowledge mingled with ignorance, producing doubt.

A great difficulty in this subject consists in acquiring a precise notion of the matter treated. What is it that we number, and measure, and calculate in the theory of probabilities? Is it belief, or opinion, or doubt, or knowledge, or chance, or necessity, or want of art? Does probability exist in the things which are probable, or in the mind which regards them as such? The etymology of the name lends us no assistance: for, curiously enough, *probable* is ultimately the same word as *provable*, a good instance of one word becoming differentiated to two opposite meanings.

Chance cannot be the subject of the theory, because there is really no such thing as chance, regarded as producing and governing events. The word chance signifies *falling*, and the notion of falling is continually used as a simile to express uncertainty, because we can seldom predict how a die, a coin, or a leaf will fall, or when a bullet will hit the mark. But everyone sees, after a little reflection, that it is in our knowledge the deficiency lies, not in the certainty of nature’s laws. There is no doubt in lightning as to the point it shall strike; in the greatest storm there is nothing capricious; not a grain of sand lies upon the beach, but infinite knowledge would account for its lying there; and the course of every falling leaf is guided by the principles of mechanics which rule the motions of the heavenly bodies.

Chance then exists not in nature, and cannot coexist with knowledge; it is merely an expression, as Laplace remarked, for our ignorance of the causes in action, and our consequent inability to predict the result, or to bring it about infallibly. In nature the happening of an event has been pre-determined from the first fashioning of the universe. *Probability belongs wholly to the mind.* This is proved by the fact that different minds may regard the very same event at the same time with widely different degrees of probability. A steam-vessel, for instance, is missing and some persons believe that she has sunk in mid-ocean; others think differently. In the event itself there can be no such uncertainty; the steam-vessel either has sunk or has not sunk, and no subsequent discussion of the probable nature of the event can alter the fact. Yet the probability of the event will really vary from day to day, and from mind to mind, according as the slightest information is gained regarding the vessels met at sea, the weather prevailing there, the signs of wreck picked up, or the previous condition of the vessel. Probability thus belongs to our mental condition, to the light in which we regard events, the occurrence or non-occurrence of which is certain in themselves. Many writers accordingly have asserted that probability is concerned with degree or quantity of belief. De Morgan says,[110] “By degree of probability we really mean or ought to mean degree of belief.” The late Professor Donkin expressed the meaning of probability as “quantity of belief;” but I have never felt satisfied with such definitions of probability. The nature of *belief* is not more clear to my mind than the notion which it is used to define. But an all-sufficient objection is, that *the theory does not measure what the belief is, but what it ought to be*. Few minds think in close accordance with the theory, and there are many cases of evidence in which the belief existing is habitually different from what it ought to be. Even if the state of belief in any mind could be measured and expressed in figures, the results would be worthless. The value of the theory consists in correcting and guiding our belief, and rendering our states of mind and consequent actions harmonious with our knowledge of exterior conditions.

[110] *Formal Logic*, p. 172.

This objection has been clearly perceived by some of those who still used quantity of belief as a definition of probability. Thus De Morgan adds--“Belief is but another name for imperfect knowledge.” Donkin has well said that the quantity of belief is “always relative to a particular state of knowledge or ignorance; but it must be observed that it is absolute in the sense of not being relative to any individual mind; since, the same information being presupposed, all minds *ought* to distribute their belief in the same way.”[111] Boole seemed to entertain a like view, when he described the theory as engaged with “the equal distribution of ignorance;”[112] but we may just as well say that it is engaged with the equal distribution of knowledge.

[111] *Philosophical Magazine*, 4th Series, vol. i. p. 355.

[112] *Transactions of the Royal Society of Edinburgh*, vol. xxi. part 4.

I prefer to dispense altogether with this obscure word belief, and to say that the theory of probability deals with *quantity of knowledge*, an expression of which a precise explanation and measure can presently be given. An event is only probable when our knowledge of it is diluted with ignorance, and exact calculation is needed to discriminate how much we do and do not know. The theory has been described by some writers as professing *to evolve knowledge out of ignorance*; but as Donkin admirably remarked, it is really “a method of avoiding the erection of belief upon ignorance.” It defines rational expectation by measuring the comparative amounts of knowledge and ignorance, and teaches us to regulate our actions with regard to future events in a way which will, in the long run, lead to the least disappointment. It is, as Laplace happily said, *good sense reduced to calculation*. This theory appears to me the noblest creation of intellect, and it passes my conception how two such men as Auguste Comte and J. S. Mill could be found depreciating it and vainly questioning its validity. To eulogise the theory ought to be as needless as to eulogise reason itself.

*Fundamental Principles of the Theory.*

The calculation of probabilities is really founded, as I conceive, upon the principle of reasoning set forth in preceding chapters. We must treat equals equally, and what we know of one case may be affirmed of every case resembling it in the necessary circumstances. The theory consists in putting similar cases on a par, and distributing equally among them whatever knowledge we possess. Throw a penny into the air, and consider what we know with regard to its way of falling. We know that it will certainly fall upon a side, so that either head or tail will be uppermost; but as to whether it will be head or tail, our knowledge is equally divided. Whatever we know concerning head, we know also concerning tail, so that we have no reason for expecting one more than the other. The least predominance of belief to either side would be irrational; it would consist in treating unequally things of which our knowledge is equal.

The theory does not require, as some writers have erroneously supposed, that we should first ascertain by experiment the equal facility of the events we are considering. So far as we can examine and measure the causes in operation, events are removed out of the sphere of probability. The theory comes into play where ignorance begins, and the knowledge we possess requires to be distributed over many cases. Nor does the theory show that the coin will fall as often on the one side as the other. It is almost impossible that this should happen, because some inequality in the form of the coin, or some uniform manner in throwing it up, is almost sure to occasion a slight preponderance in one direction. But as we do not previously know in which way a preponderance will exist, we have no reason for expecting head more than tail. Our state of knowledge will be changed should we throw up the coin many times and register the results. Every throw gives us some slight information as to the probable tendency of the coin, and in subsequent calculations we must take this into account. In other cases experience might show that we had been entirely mistaken; we might expect that a die would fall as often on each of the six sides as on each other side in the long run; trial might show that the die was a loaded one, and falls most often on a particular face. The theory would not have misled us: it treated correctly the information we had, which is all that any theory can do.

It may be asked, as Mill asks, Why spend so much trouble in calculating from imperfect data, when a little trouble would enable us to render a conclusion certain by actual trial? Why calculate the probability of a measurement being correct, when we can try whether it is correct? But I shall fully point out in later parts of this work that in measurement we never can attain perfect coincidence. Two measurements of the same base line in a survey may show a difference of some inches, and there may be no means of knowing which is the better result. A third measurement would probably agree with neither. To select any one of the measurements, would imply that we knew it to be the most nearly correct one, which we do not. In this state of ignorance, the only guide is the theory of probability, which proves that in the long run the mean of divergent results will come most nearly to the truth. In all other scientific operations whatsoever, perfect knowledge is impossible, and when we have exhausted all our instrumental means in the attainment of truth, there is a margin of error which can only be safely treated by the principles of probability.

The method which we employ in the theory consists in calculating the number of all the cases or events concerning which our knowledge is equal. If we have the slightest reason for suspecting that one event is more likely to occur than another, we should take this knowledge into account. This being done, we must determine the whole number of events which are, so far as we know, equally likely. Thus, if we have no reason for supposing that a penny will fall more often one way than another, there are two cases, head and tail, equally likely. But if from trial or otherwise we know, or think we know, that of 100 throws 55 will give tail, then the probability is measured by the ratio of 55 to 100.

The mathematical formulæ of the theory are exactly the same as those of the theory of combinations. In this latter theory we determine in how many ways events may be joined together, and we now proceed to use this knowledge in calculating the number of ways in which a certain event may come about. It is the comparative numbers of ways in which events can happen which measure their comparative probabilities. If we throw three pennies into the air, what is the probability that two of them will fall tail uppermost? This amounts to asking in how many possible ways can we select two tails out of three, compared with the whole number of ways in which the coins can be placed. Now, the fourth line of the Arithmetical Triangle (p. 184) gives us the answer. The whole number of ways in which we can select or leave three things is eight, and the possible combinations of two things at a time is three; hence the probability of two tails is the ratio of three to eight. From the numbers in the triangle we may similarly draw all the following probabilities:--

One combination gives 0 tail. Probability 1/8. Three combinations gives 1 tail. Probability 3/8. Three combinations give 2 tails. Probability 3/8. One combination gives 3 tails. Probability 1/8.

We can apply the same considerations to the imaginary causes of the difference of stature, the combinations of which were shown in p. 188. There are altogether 128 ways in which seven causes can be present or absent. Now, twenty-one of these combinations give an addition of two inches, so that the probability of a person under the circumstances being five feet two inches is 21/128. The probability of five feet three inches is 35/128; of five feet one inch 7/128; of five feet 1/128, and so on. Thus the eighth line of the Arithmetical Triangle gives all the probabilities arising out of the combinations of seven causes.

*Rules for the Calculation of Probabilities.*

I will now explain as simply as possible the rules for calculating probabilities. The principal rule is as follows:--

Calculate the number of events which may happen independently of each other, and which, as far as is known, are equally probable. Make this number the denominator of a fraction, and take for the numerator the number of such events as imply or constitute the happening of the event, whose probability is required.

Thus, if the letters of the word *Roma* be thrown down casually in a row, what is the probability that they will form a significant Latin word? The possible arrangements of four letters are 4 × 3 × 2 × 1, or 24 in number (p. 178), and if all the arrangements be examined, seven of these will be found to have meaning, namely *Roma*, *ramo*, *oram*, *mora*, *maro*, *armo*, and *amor*. Hence the probability of a significant result is 7/24.

We must distinguish comparative from absolute probabilities. In drawing a card casually from a pack, there is no reason to expect any one card more than any other. Now, there are four kings and four queens in a pack, so that there are just as many ways of drawing one as the other, and the probabilities are equal. But there are thirteen diamonds, so that the probability of a king is to that of a diamond as four to thirteen. Thus the probabilities of each are proportional to their respective numbers of ways of happening. Again, I can draw a king in four ways, and not draw one in forty-eight, so that the probabilities are in this proportion, or, as is commonly said, the *odds* against drawing a king are forty-eight to four. The odds are seven to seventeen in favour, or seventeen to seven against the letters R,o,m,a, accidentally forming a significant word. The odds are five to three against two tails appearing in three throws of a penny. Conversely, when the odds of an event are given, and the probability is required, *take the odds in favour of the event for numerator, and the sum of the odds for denominator*.

It is obvious that an event is certain when all the combinations of causes which can take place produce that event. If we represent the probability of such event according to our rule, it gives the ratio of some number to itself, or unity. An event is certain not to happen when no possible combination of causes gives the event, and the ratio by the same rule becomes that of 0 to some number. Hence it follows that in the theory of probability certainty is expressed by 1, and impossibility by 0; but no mystical meaning should be attached to these symbols, as they merely express the fact that *all* or *no* possible combinations give the event.

By a *compound event*, we mean an event which may be decomposed into two or more simpler events. Thus the firing of a gun may be decomposed 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 1/2 × 1/2, or 1/4; the probability of throwing it three times running is 1/2 × 1/2 × 1/2, or 1/8; a result agreeing with that obtained in an apparently different manner (p. 202). 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, since the probability of throwing head at the first trial is 1/2, and at the second trial also 1/2, the probability of throwing it in the first two throws is 1/2 + 1/2, or certainty. Not only is this result evidently absurd, but a repetition of the process would lead us to a probability of 1-1/2 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 1/2 + 1/2 × 1/2 or 3/4, 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, α, β, γ, then the negatives of those events, expressing the absence of the events, will have the probabilities 1 - α, 1 - β, 1 - γ. 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 αβγ; of A*bc*, α(1 - β)(1 - γ).

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 ꖌ  B into AB ꖌ A*b* ꖌ *a*B, and substituting α and β, the probabilities of A and B respectively, we obtain α . β + α . (1 - β) + (1 - α) . β or α + β - α . β. 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 A*b* ꖌ *a*B. Now if we take μ = probability of A*b* and ν = probability of *a*B, then we may add simply, and the probability of A*b* ꖌ *a*B is μ + ν.

Let the reader carefully observe that if the combination AB cannot exist, the probability of A*b* 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,[113] Boole obtained his results by an arbitrary assumption, which is only the most probable, and not the only possible assumption. 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,[114] 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.

[113] *Philosophical Magazine*, 4th Series, vol. vii. p. 465; vol. viii. p. 91.

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

*Comparison of the Theory with Experience.*

The Laws of Probability rest upon the fundamental 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 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 1·02.[115]

[115] *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 table:--

+-------------------+-----------+---------+---------+----------+-----------+ |Character of Throw.|Theoretical| First | Second | Average. |Divergence.| | | Numbers. | Series. | Series. | | | +-------------------+-----------+---------+---------+----------+-----------+ | 10 Heads 0 Tail | 1 | 3 | 1 | 2 | + 1 | | 9 " 1 " | 10 | 12 | 23 | 17-1/2 | + 7-1/2 | | 8 " 2 " | 45 | 57 | 73 | 65 | + 20 | | 7 " 3 " | 120 | 129 | 123 | 126 | + 6 | | 6 " 4 " | 210 | 181 | 190 | 185-1/2 | - 25 | | 5 " 5 " | 252 | 257 | 232 | 244-1/2 | - 7-1/2 | | 4 " 6 " | 210 | 201 | 197 | 199 | - 11 | | 3 " 7 " | 120 | 111 | 119 | 115 | - 5 | | 2 " 8 " | 45 | 52 | 50 | 51 | + 6 | | 1 " 9 " | 10 | 21 | 15 | 18 | + 8 | | 0 " 10 " | 1 | 0 | 1 | 1/2 | - 1/2 | +-------------------+-----------+---------+---------+----------+-----------+ | Totals | 1024 | 1024 | 1024 | 1024 | - 1 | +-------------------+-----------+---------+---------+----------+-----------+

The whole number of single throws of coins amounted to 10 × 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 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,[116] 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.

[116] *Encyclopædia Metropolitana*, art. *Probabilities*, p. 396.

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 1/2 that A is B, and also 1/2 that B is C, the conclusion that A is C, on the ground of these premises, is 1/2 × 1/2 or 1/4. Similarly if there be any number of premises requisite to the establishment of a conclusion and their probabilities be *p*, *q*, *r*, &c., the probability of the conclusion on the ground of these premises is *p* × *q* × *r* × ... This product has but a small value, unless each of the quantities *p*, *q*, &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[117] 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[118] 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.” 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. As a rope does not necessarily break because one or two strands in it fail, 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 1/2.

[117] *Elements of Logic*, Book III. sections 11 and 18.

[118] *Encyclopædia Metropolitana*, art. *Probabilities*, p. 400.

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 1/2. If we also have the probabilities that A is B, 1/2 and that B is C, 1/2 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 1/2 that we shall not require the special argument; a probability 1/2 that we shall, and a probability 1/4 that the argument does in that case establish it. Thus the complete result is 1/2 + 1/2 × 1/4, or 5/8. In general language, if *a* be the probability founded on a particular argument, and *c* the antecedent probability of the event, the general result is 1 - (1 - *a*)(1 - *c*), or *a* + *c* - *ac*.

We may put it still more generally in this way:--Let *a*, *b*, *c*, &c. be the probabilities of a conclusion grounded on various arguments. 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 is (1 - *a*)(1 - *b*)(1 - *c*) ...; therefore the probability that the conclusion will not fail is 1 - (1 - *a*)(1 - *b*)(1 - *c*) ... &c. It follows that every argument in favour of a conclusion, 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 or to the supposed law of nature, and every reasoner may be assumed to be dealing plainly, and putting forward the whole force of evidence which he possesses in its favour. If he brings but one argument, and its probability *a* is small, then in the formula 1 - (1 - *a*)(1 - *c*) both *a* and *c* are small, and the whole expression has but little value. The whole effect of an argument thus turns upon the question whether other arguments remain, so that we can introduce other factors (1 - *b*), (1 - *d*), &c., into the above expression. In a court of justice, in a publication having an express purpose, and in many other cases, it is doubtless right to assume that the whole evidence considered to have any value as regards the conclusion asserted, is put forward.

To assign the antecedent probability of any proposition, may be a matter of difficulty or impossibility, and one with which logic and the theory of probability have little concern. From the general body of science in our possession, we must in each case make the best judgment we can. But in the absence of all knowledge the probability should be considered = 1/2, for if we make it less than this we incline to believe it false rather than true. Thus, before we possessed any means of estimating the magnitudes of the fixed stars, the statement that Sirius was greater than the sun had a probability of exactly 1/2; it was as likely that it would be greater as that it would be smaller; and so of any other star. This was the assumption which Michell made in his admirable speculations.[119] It might seem, indeed, that as every proposition expresses an agreement, and the agreements or resemblances between phenomena are infinitely fewer than the differences (p. 44), every proposition should in the absence of other information be infinitely improbable. But in our logical system every term may be indifferently positive or negative, so that we express under the same form as many differences as agreements. It is impossible therefore that we should have any reason to disbelieve rather than to believe a statement about things of which we know nothing. We can hardly indeed invent a proposition concerning the truth of which we are absolutely ignorant, except when we are entirely ignorant of the terms used. If I ask the reader to assign the odds that a “Platythliptic Coefficient is positive” he will hardly see his way to doing so, unless he regard them as even.

[119] *Philosophical Transactions* (1767). Abridg. vol. xii. p. 435.

The assumption that complete doubt is properly expressed by 1/2 has been called in question by Bishop Terrot,[120] who proposes instead the indefinite symbol 0/0; and he considers that “the *à priori* probability derived from absolute ignorance has no effect upon the force of a subsequently admitted probability.” But if we grant that the probability may have any value between 0 and 1, and that every separate value is equally likely, then *n* and 1 - *n* are equally likely, and the average is always 1/2. Or we may take *p* . *dp* to express the probability that our estimate concerning any proposition should lie between *p* and *p* + *dp*. The complete probability of the proposition is then the integral taken between the limits 1 and 0, or again 1/2.

[120] *Transactions of the Edinburgh Philosophical Society*, vol. xxi. p. 375.

*Difficulties of the Theory.*

The theory of probability, though undoubtedly true, requires very careful application. Not only is it a branch of mathematics in which oversights are frequently committed, but it is a matter of great difficulty in many cases, to be sure that the formula correctly represents the data of the problem. These difficulties often arise from the logical complexity of the conditions, which might be, perhaps, to some extent cleared up by constantly bearing in mind the system of combinations as developed in the Indirect Logical Method. In the study of probabilities, mathematicians had unconsciously employed logical processes far in advance of those in possession of logicians, and the Indirect Method is but the full statement of these processes.

It is very curious how often the most acute and powerful intellects have gone astray in the calculation of probabilities. Seldom was Pascal mistaken, yet he inaugurated the science with a mistaken solution.[121] Leibnitz fell into the extraordinary blunder of thinking that the number twelve was as probable a result in the throwing of two dice as the number eleven.[122] In not a few cases the false solution first obtained seems more plausible to the present day than the correct one since demonstrated. James Bernoulli candidly records two false solutions of a problem which he at first thought self-evident; and he adds a warning against the risk of error, especially when we attempt to reason on this subject without a rigid adherence to methodical rules and symbols. Montmort was not free from similar mistakes. D’Alembert constantly fell into blunders, and could not perceive, for instance, that the probabilities would be the same when coins are thrown successively as when thrown simultaneously. Some men of great reputation, such as Ancillon, Moses Mendelssohn, Garve, Auguste Comte,[123] Poinsot, and J. S. Mill,[124] have so far misapprehended the theory, as to question its value or even to dispute its validity. The erroneous statements about the theory given in the earlier editions of Mill’s *System of Logic* were partially withdrawn in the later editions.

[121] Montucla, *Histoire des Mathématiques*, vol. iii. p. 386.

[122] Leibnitz *Opera*, Dutens’ Edition, vol. vi. part i. p. 217. Todhunter’s *History of the Theory of Probability*, p. 48. To the latter work I am indebted for many of the statements in the text.

[123] *Positive Philosophy*, translated by Martineau, vol. ii. p. 120.

[124] *System of Logic*, bk. iii. chap. 18, 5th Ed. vol. ii. p. 61.

Many persons have a fallacious tendency to believe that when a chance event has happened several times together in an unusual conjunction, it is less likely to happen again. D’Alembert seriously held that if head was thrown three times running with a coin, tail would more probably appear at the next trial.[125] Bequelin adopted the same opinion, and yet there is no reason for it whatever. If the event be really casual, what has gone before cannot in the slightest degree influence it. As a matter of fact, the more often a casual event takes place the more likely it is to happen again; because there is some slight empirical evidence of a tendency. The source of the fallacy is to be found entirely in the feelings of surprise with which we witness an event happening by chance, in a manner which seems to proceed from design.

[125] Montucla, *Histoire*, vol. iii. p. 405; Todhunter, p. 263.

Misapprehension may also arise from overlooking the difference between permutations and combinations. To throw ten heads in succession with a coin is no more unlikely than to throw any other particular succession of heads and tails, but it is much less likely than five heads and five tails without regard to their order, because there are no less than 252 different particular throws which will give this result, when we abstract the difference of order.

Difficulties arise in the application of the theory from our habitual disregard of slight probabilities. We are obliged practically to accept truths as certain which are nearly so, because it ceases to be worth while to calculate the difference. No punishment could be inflicted if absolutely certain evidence of guilt were required, and as Locke remarks, “He that will not stir till he infallibly knows the business he goes about will succeed, will have but little else to do but to sit still and perish.”[126] There is not a moment of our lives when we do not lie under a slight danger of death, or some most terrible fate. There is not a single action of eating, drinking, sitting down, or standing up, which has not proved fatal to some person. Several philosophers have tried to assign the limit of the probabilities which we regard as zero; Buffon named 1/10,000, because it is the probability, practically disregarded, that a man of 56 years of age will die the next day. Pascal remarked that a man would be esteemed a fool for hesitating to accept death when three dice gave sixes twenty times running, if his reward in case of a different result was to be a crown; but as the chance of death in question is only 1 ÷ 6^{60}, or unity divided by a number of 47 places of figures, we may be said to incur greater risks every day for less motives. There is far greater risk of death, for instance, in a game of cricket or a visit to the rink.

[126] *Essay concerning Human Understanding*, bk. iv. ch. 14. § 1.

Nothing is more requisite than to distinguish carefully between the truth of a theory and the truthful application of the theory to actual circumstances. As a general rule, events in nature and art will present a complexity of relations exceeding our powers of treatment. The intricate action of the mind often intervenes and renders complete analysis hopeless. If, for instance, the probability that a marksman shall hit the target in a single shot be 1 in 10, we might seem to have no difficulty in calculating the probability of any succession of hits; thus the probability of three successive hits would be one in a thousand. But, in reality, the confidence and experience derived from the first successful shot would render a second success more probable. The events are not really independent, and there would generally be a far greater preponderance of runs of apparent luck, than a simple calculation of probabilities could account for. In some persons, however, a remarkable series of successes will produce a degree of excitement rendering continued success almost impossible.

Attempts to apply the theory of probability to the results of judicial proceedings have proved of little value, simply because the conditions are far too intricate. As Laplace said, “Tant de passions, d’intérêts divers et de circonstances compliquent les questions relatives à ces objets, qu’elles sont presque toujours insolubles.” Men acting on a jury, or giving evidence before a court, are subject to so many complex influences that no mathematical formulas can be framed to express the real conditions. Jurymen or even judges on the bench cannot be regarded as acting independently, with a definite probability in favour of each delivering a correct judgment. Each man of the jury is more or less influenced by the opinion of the others, and there are subtle effects of character and manner and strength of mind which defy analysis. Even in physical science we can in comparatively few cases apply the theory in a definite manner, because the data required are too complicated and difficult to obtain. But such failures in no way diminish the truth and beauty of the theory itself; in reality there is no branch of science in which our symbols can cope with the complexity of Nature. As Donkin said,--

“I do not see on what ground it can be doubted that every definite state of belief concerning a proposed hypothesis, is in itself capable of being represented by a numerical expression, however difficult or impracticable it may be to ascertain its actual value. It would be very difficult to estimate in numbers the *vis viva* of all the particles of a human body at any instant; but no one doubts that it is capable of numerical expression.”[127]

[127] *Philosophical Magazine*, 4th Series, vol. i. p. 354.

The difficulty, in short, is merely relative to our knowledge and skill, and is not absolute or inherent in the subject. We must distinguish between what is theoretically conceivable and what is practicable with our present mental resources. Provided that our aspirations are pointed in a right direction, we must not allow them to be damped by the consideration that they pass beyond what can now be turned to immediate use. In spite of its immense difficulties of application, and the aspersions which have been mistakenly cast upon it, the theory of probabilities, I repeat, is the noblest, as it will in course of time prove, perhaps the most fruitful branch of mathematical science. It is the very guide of life, and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of probabilities. In the next chapter we proceed to consider how the whole cogency of inductive reasoning rests upon probabilities. The truth or untruth of a natural law, when carefully investigated, resolves itself into a high or low degree of probability, and this is the case whether or not we are capable of producing precise numerical data.