turn those which precede to account, in examining opinions of various kinds, whether on this subject at large, or on particular cases of its application.
The doctrine of probabilities seems to some to assume a sort of power of prophesying, or of predicting the run of events ; to others, it appears that unless such a power of prophesying be attained, the theory can be of no use. Both notions are correct in one sense and in-correct in another: there is prophecy, but not of parevents, and derived, not from inspiration, but from observation. The astronomer predicts—and all the world knows that his predictions daily come true. His means of prophecy are aided by deduction from certain notions of which, be the cause what it may, we are as certain as of our own existence. From his very distinct (and therefore often called intuitive) perception, that two straight lines cannot inclose a space, and various other axioms of arithmetic and geometry, he is able to make his observations tell him more as to the future motions of our system than his unassisted perceptions of the past could ever have accomplished. He is a dealer in probabilities of a very high order. But before his prediction appears, it is necessary that he should consider much more doubtful questions of proThe minute errors of observation, coupled with the various trifling effects which result from yet uncauses, oblige him to have recourse to the principles which we have explained in the preceding chapters.
Again, there is no prophecy of particular events in the theory of probabilities, of which it is the very esthat there should be more or less tendency to falsehood in every one of its assertions. No result is announced, except as having a certain chance in its favour, which implies also a certain chance against it.
With regard to the second class of assertions, namely, that unless the theory of probabilities enable us to predict, it can be of no use — it may be said that, for the purpose contemplated, it is of no use. Theory would

I

never enable us to tell what face of a die will be turned up in any one instance, nor would the maxims of our science be worth putting into practice with respect to an event which is to happen only a few times. If a man were determined to run six hazards, and never to gamble afterwards, say if he were determined to wager twice upon a pair of dice giving doublets, I should think, it perfectly immaterial whether he accepted an even wager, namely 5 to 1, or not. For though, in the long run, only one throw out of six will give doublets, yet the probability that six throws will give such a pair once at least is not very great. It is as a provider of general rules of conduct that the science is valuable ; the adherence to rules being desirable on precisely the same principles as those which obtain in morals or legisno maxim of which will be found to meet every case which will occur.
It is an assumption of this theory that nothing ever did happen, or ever will happen, without some particular reason why it should have been precisely what it was, and not any thing else. Conceive it possible that a ball which is white might have been black, without the alteration of any action or circumstance which took place in time previous to the moment at which the ball is shown, and the foundations of the theory of probabilities have ceased to exist in the mind which has formed that conception. There is no one but will admit, that out of a box, which contains nothing but two black balls, nothing but black balls can be drawn ; and that out of a box which contains only two white balls, no black balls can be drawn. The difficulty lies in a clear perception of the remaining assertion ; namely, that when the box contains one white ball and one black ball, a very large number of drawings will give as many white as black nearly, and the more nearly the greater the number. This proposition might be proved in three ways : firstly, by actual experiment ; secondly, by showing that out of all the possible cases which can happen, those in which black and white are equal, or

nearly equal, much exceed in number all the rest put together ; thirdly, by showing that there can be no possible reason for an excess of white, which does not equally, by express condition of the question, apply in faof an excess of black. The last is more unanswerthan convincing ; the second really shows that the event which we propose and treat as one event, namely, " as many white as black, or nearly so,^{" is, in fact, a collection of a large number of events, much exceeding in number all the rest which can happen. It is as if, having a million of possible cases, I separated nine hundred thousand from the rest, called each of them A, and each of the rest B, and then asserted that A would happen more often than B. But, nevertheless, I susthat to the first mode of demonstration, actual experiment, most persons owe that degree of confidence in the theory, which (often without knowing it) they exhibit in the affairs of life; and I derive such a suspicion from observing that every result of the theory of probabilities which is not of a nature to admit of every-day confirmation, or which would escape an inattentive observer, is looked upon with distrust. In no case is this more obvious than in the prevailing notions with regard to luck.
It is observed that some people always have luck at cards. The order of things seems disturbed at their caprice; if they sit opposite to the dealer at whist, then there is always an undue proportion of trumps among the cards which come second, sixth, tenth, &c., up to the fiftieth; while, when they become before the dealer — Hocus Pocus (for to no other spirit, ancient or modern, can the agency be attributed) puts all the good cards, third, seventh, eleventh, &c. The fact is stated as a sort of mystery, and we hear of people who are always lucky at cards and never at dice, or vice versa. The statement implies that the parties who make it believe there is something in luck— an asserwhich I do not think of questioning ; for, as I}

1 2

shall proceed to show, it would be the most improbable thing imaginable that there should be no such lucky people.

Firstly, every question of probabilities stands in precisely the same relation to our faculties, whether we suppose a moral government of the universe, or none at all, provided that we have no reason to suppose we know any thing of the plan of that government in the particular case in question. If I am before an urn which contains a black and a white ball, which is all I know, I am then disposed to say the chances are even. The ball which I am to draw is undetermined (by me), and that which we call chance appears to exist. But suppose I draw the ball, and without looking at it hold it in my hand. That which we call chance has ceased to exist — the ball is actually determined, and I am clearly and physically placed in the same position as I should have had before the drawing, if a superintending power, capable of guiding my thoughts and actions without my perceiving it, had predetermined which I should draw. But my position with respect to knowledge of the ball is not in any way changed, either by the predetermination of the superintending power before the drawing or by my own act of drawing, as long as 1 do not know what I am to draw or have drawn. It tells me nothing, if I hear that the drawing is settled, unless it be in a manner by which I can form some guess as to the nature of the settlement. Consequently we must not, unless some reason be shown for it, consider the question of the luck of individuals in any other light, with reference to calculation, than that in which it would have been placed by the supposition that, all imaginable species of fortune being described on the tickets of a lottery, each individual had one drawing made for him at birth, which should describe his future successes and reverses. To create an analogous question, within reasonable numerical limits, let us suppose a thousand individuals, each of whom is to play two thousand deals at

ON COMMON NOTIONS OF PROBABILITY.    117

whist, with a given suit as trumps.* Let there be a lottery, containing an enormous number of books, in each of which 2000 deals are described, and let the books be so many in number that among them is one containing every possible set of 2000 deals which can be imagined, the four hands in each deal being described, and that allotted to the drawer of the book being marked as such. Let each individual draw one of these books, replacing it before the next drawer arrives: these individuals are then precisely in the same situation with regard to us, if their hands are to be dealt to them according to the directions laid down in their books, as if the distribution were made by accident (as we call it). Now the question is, which is most likely, that the luck of these individuals shall all be nearly the same, or that some of them shall have a marked predominance over others? To take one simple question : consider only the chance of gaining the ace of trumps. Excluding the dealer^{'s advantage, to simplify the question, the chance of any one individual gaining the ace of trumps at any given deal is }. Considering 2000 deals, he has a very good chance of gaining it 500 times, or a few more or less. But the probabilities are much in favour of several of these 1000 individuals having a very different lot from the average. Frame a set of cirin this respect against which it shall be twenty to one, and (page 43) it is a hundred to one that this fate (or a better) shall be found to be that of some one or other out of any 92 individuals taken at hazard from among the thousand. And when to the chance of holding the ace of trumps we add the various others which constitute a good hand at the game, we thereby much increase the probability of large fluctuations, one way or the other; and though it is certain that uniformity will be found in the average lot of a large number of per}

sons, yet the larger the number, the greater will be the extremes of fluctuation.
Now it must be noticed, that this variation is the thing observed — not on one side only, but on both. For every one who is lucky at cards, there is another who is unlucky. It would be, indeed, such a sort of mystery as that which I am endeavouring to explain, if the exto common luck were all on one side, or if there were no such thing at all as uncommon luck, or only in very few instances. This latter would be the same sort of phenomenon as we should see if a halfpenny gave head and tail alternately through an enormous recurrence of throws. The event observed is precisely that which might have been expected beforehand. If by thinking mysteriously of the fluctuations of luck which are observed in comparing the fortunes of individuals, any reader should mean to imply that the alternative, namely, slight individual departure from the average, would not have been mysterious, he is in a singular error. The state of things which he would regard with no wonder would be an apparent interference with the material world on the part of its governor, without the intermediate agency of any second causes ; that is, something resembling a miracle. For though the phiin such a case, would suspect an intermediate cause, and endeavour to discover it, this consideration does not enter into the view of people in general. 'When the world wonders, whether at one side or another of a question of probabilities, it is at the want of ally apphysical or moral reason : on which account they refer it to the Creator in a manner different from that in which they refer what they call usual occurrences.
The law of individual cases is, that there shall be marked differences ; of the masses, that there shall be great approach to uniformity. There are a hundred years in which, and hundreds of diseases by which, any individual who is born may die : a lottery, which should contain one ticket for every disorder, repeated as often as there are years of age in which it has been
ON COMMON NOTIONS OF PROBABILITY.    119
fatal, would present at least 20,000 chances. Before an individual is born,it is, say 20,000 to 1 against his dying at a given age of a given malady ; and yet, even with such imperfect observations as exist at present, it begins to be seen that uniformity is the law of large masses compared with each other. I will illustrate this by some cases. Few things appear more varied than the distribution of maladies, that of criminal acts, and that of the sex of children in different families. I have taken purposely a case of evil, physical and moral, and one which is neither. The experience of any one individual might lead him to say that it is no uncommon thing for three or four times as many persons to die of consumption in one period of five years as in the previous period ; but the experience of one large city will show that such is not the case. The bills of morin London showed the following results; the upper line denoting the last year of the five in question, and the lower line the average number in every thousand deaths which were caused by consumption, or what was called such."
1732    1737    1742    1747    1752    1757
135    163    165    180    187    197.
Here is nothing like enormous fluctuation. The gradual increase of the number shows an increasing tendency to the complaints then described under the head consumption, but cannot be called fluctuation, being itself regular.
The number of murders committed in the whole extent of any one country might be supposed liable to very large yearly fluctuation, and still more the comparative numbers committed with different classes of weapons. A few years ago, extreme derision would have followed the assertion that the sword and pistol would be felo
The known loose manner in which these Tables were put together does not affect the argument, further than to favour its conclusion. The chances of error in the description increase the probability of fluctuation in the results.

piously used in different years by nearly the same num^{bers. Let us look at the following Table, extracted from the Essai de Physique Sociale of M. Quetelet. The country referred to is France.}
    

I now compare the number of male and female bapregistered in England in 1821, and the nine folyears. For these successive years it is found that for 1000 girls baptised there were 1048, 1047, 1047, 1041, 1049, 1046, 1047, 1043, 1043, and 1034 boys.
Such cases as the preceding tend to establish the law in question ; namely, that different large masses of facts, collected under the same circumstances, will present nearly the same averages. I now proceed to another point.
When two circumstances happen to change together, it is frequently presumed that they are connected with each other, when, in truth, there is no reason for any such supposition. In order to justify any notion of

necessary connection it is necessary that the two cirshould always happen together, and that one should never happen without the other. If it should only be observed that one is very frequently accompanied by the other, we must then inquire into the probability that either may happen without the other. If two events are almost always happening, then it is evident that very frequent coincidence is no evidence of connecso long as exceptions tell us that there is no necesconnection. And even if we always observe A to be immediately followed by B, it does not immediately follow that they are necessarily connected. We must remember that the phenomenon is this — our perception of A is immediately followed by our perception of B. This may arise in different ways, as follows.