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.12 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.13 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,14 Poinsot, and J. S. Mill,15 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 The erroneous statements about the theory given in the earlier editions of Mill’s System of Logic were partially withdrawn in the later editions.
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.16 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.
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.”17 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 ÷ 660, 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.
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