PROFESSOR DE MORGANS ESSAY ON PROBABILITY.
THIS Essay is an object of interest to the writer of the present article, as containing the substance of a contest held nearly forty years ago, under circumstances of immense disparity of age and powers, with the most eminent mathematician of that day in Cam- bridge; which ended in the senior refusing to concede, and the junior to believe. The rebellious catechumen declared to be unintel- ligible, the reasoning propounded in the beginning of Woon's chapter on Chances; and avowed his heretical belief, that the whole was reducible to the mere question of the number of ways in which an assigned future event could happen or fail. And now he has the satisfaction of seeing a ripe philosopher advance to support his juvenile heterodoxy. When we speculate on what is popularly termed the "chance" of a particular event's happening or not happening,—as, for in- stance, to take one of the simplest cases in nature, the " chance " of throwing an ace with a single die,—the only point about the case capable of reduction to an accurate standard, is the fact that if the throws be continued to a very great or what is called an un- limited extent, the number of aces thrown will approach to one- sixth of the whole. If on trial it should approach to anything else, as to a fifth or to a seventh, this would be evidence before any jury in the world that the dice were loaded, pr that there did not exist that equable tendency to throw any of the six possible numbers, which was professed. What we want to reduce to calculation then—and in truth, all we can reduce to calculation if we try—is the ratio between the hits and the misses which will occur, or at all events will be very nearly approximated to, if the number of throws be increased to an unlimited extent. If the question is anything else, its solution must depend on elements not reducible to numerical representation, or at most must have only a dim and distant reference to the attainable principle which has been described. If a con- demned criminal, for example, be promised his life if he throw an ace,— whether he shall be saved or not appears utterly incapable of being reduced to numerical calculation. Either he will be saved entirely or be lost entirely ; there is no medium. What we know is, that if there were, for instance, three hundred such criminals brought out to throw for their lives in succession, fifty of them, more or less, would escape, and the rest be condemned. And thence we gather a further conviction, that the single criminal who shall be supposed to be really con- cerned, is in a very dangerous way,—that it is much more likely he will be lost than saved ; and so strongly does this inference exist, that where an individual has been saved in spite of an extraordi- nary accumulation of unfavourable circumstances, as for instance in the reported cases (if any are true) where a man's life has been saved by throwing an ace half a dozen times over in succession, mankind have invariably proceeded to the conclusion that it was a miraculous declaration of innocence, on the principle of the ordeal of our forefathers. And if we ask ourselves what is meant by the" ex- traordinary accumulation of unfavourable circumstances," under which that an individual should escape, is considered as miracu- lous; it means this,—that he has escaped under circumstances such that, had forty thousand men been exposed to the same risk, one only out of them all could have been reasonably expected to escape, and to a certainty it would not have happened to more than two or three.
The conclusion from the whole, therefore is, that when a reason- able man asks what is the chalice of an assigned future event happening or failing, he means to ask what is the proportion be- tween the successes and the failures which would take place in an unlimited number of trials under the same predisposing causes ; and if he means to ask anything else, he is not a reason- able man. For example, suppose that before the world was cir- cumnavigated, a man had insisted on asking what was the chance of its performance. He might have argued, that to circumnavi- gate the world, might be likened to six India voyages in succes- sion; and therefore if he could discover what number of ships had made six India voyages in succession, and what had failed, he would have arrived at a first approximation to the desired result. But lie must add the consideration, that after an India voyage a ship repairs losses in a known port ; and here there must be three or four such voyages or. the back of one another, without any such assistance. But this also might perhaps to a certain extent be reduced to calculation. Remained then, the danger of the un- known; as for instance, whether there might be a meridian of whirlpool passing through the antipodes, expressly intended to swallow up profane adventurers ; or a serpent of the ocean, spe- cially commissioned to guard his native deeps. On the chance of all this, nobody could say anything till it had been tried; the cir- cumnavigator therefore must have had the ces circa pectus to set all this last class of' perils at defiance. But if there are classes of future events where we predict the effects from consideration of the causes, there are others where we gather the causes from what we see of the effects. Such, for instance, are the speculations on the duration of life, or the pro- portion of the sexes ; speculations reducible to what may be de- nominated perfect accuracy in the gross, though not pretending to the smallest power in an individual case. If the actuary of an insurance-office is asked what is the chance of the young man in health and vigour who comes of age to-day, living till to-morrow, all he can answer is, that if a million of individuals under the same circumstances will offer him a very small fraction of a farthing, to return a hundred pounds to their relations in the event of death, he will with perfect confidence accept the bargain. Yet in no individual case can it be certain that there are not the seeds of death which must operate before to-morrow's noon. It is uncertain in the individual ; but it is quite certain that it is not so in many individuals ; and thus the actuary is safe.
A part of the subject, which Mr. DE Moaossr has not over- looked, is its religious or moral bearing. The dread of knowledge is far from being extinct. There are many to this day besides Mohammedans, who have an indistinct fear of poring into the awful subjects of life and death, very analogous to what our fore- fathers entertained on the pursuit of what they held forbidden sciences. The great answer in both cases, seems to be, that whatever it is given to man to discover, man may know; no proof having ever been given of the existence of any bar separating the knowable from the lawful. Whether a given man aged twenty shall live fifty years, it is not permitted to us to predict. That of a million men aged twenty some will live fifty years, is a truth as entirely within our attainment, and therefore as innocently to be written down and acted on, as that after winter comes spring, or after seed-time harvest.
The moral effects, also of an acquaintance with the mathemati- cal principles of chances, appear on the whole to be favourable. It goes to counteract the propensity to recklessness, which is at the bottom of half the vice and much of the misery of mankind. It very often, too, inserts positive lessons of caution, most useful in actual life. The distinguished individual before alluded to, used to say, "I love to show these things to our young men. See now, for instance, bow the chances against an event which is de- pendent on a succession of good fortune, accumulate. An admiral is to sail to a given port "—[it was in the admiralling days] —" where it is two to one he meets an expected reinforcement; then to the West Indies, where it is two to one he meets an enemy's fleet; and then it is two to one he beats it. What is the ultimate chance of this event? Eight out of twenty-seven, or not far from two and a half to one, against it." The tendency of these remarks, is to direct many besides actu aries, to the inspection of Mr. DE MORGAN'S Essay on Proba bilities.
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