2. To find the, probability that an event which has not hitherto failed will not fail for a certain number of new occasions, divide the number of times the event has happened increased by one, by the same number increased by one and the number of times it is to happen.
An event having happened m times without fail, the probability that it will happen n more times is m + 1/m + n + 1. Thus the probability that three new planets would obey Bode’s law is 10/13; but it must be allowed that this, as well as the previous result, would be much weakened by the fact that Neptune can barely be said to obey the law.
3. An event having happened and failed a certain number of times, to find the probability that it will happen the next time, divide the number of times the event has happened increased by one, by the whole number of times the event has happened or failed increased by two.
If an event has happened m times and failed n times, the probability that it will happen on the next occasion is m + 1/m + n + 2. Thus, if we assume that of the elements discovered up to the year 1873, 50 are metallic and 14 non-metallic, then the probability that the next element discovered will be metallic is 51/66. Again, since of 37 metals which have been sufficiently examined only four, namely, sodium, potassium, lanthanum, and lithium, are of less density than water, the probability that the next metal examined or discovered will be less dense than water is 4 + 1/37 + 2 or 5/39.
We may state the results of the method in a more general manner thus,[169]—If under given circumstances certain events A, B, C, &c., have happened respectively m, n, p, &c., times, and one or other of these events must happen, then the probabilities of these events are proportional to m + 1, n + 1, p + 1, &c., so that the probability of A will be m + 1/m + 1 + n + 1 + p + 1 + &c. But if new events may happen in addition to those which have been observed, we must assign unity for the probability of such new event. The odds then become 1 for a new event, m + 1 for A, n + 1 for B, and so on, and the absolute probability of A is m + 1/1 + m + 1 + n + 1 + &c.
It is interesting to trace out the variations of probability according to these rules. The first time a casual event happens it is 2 to 1 that it will happen again; if it does happen it is 3 to 1 that it will happen a third time; and on successive occasions of the like kind the odds become 4, 5, 6, &c., to 1. The odds of course will be discriminated from the probabilities which are successively 2/3, 3/4, 4/5, &c. Thus on the first occasion on which a person sees a shark, and notices that it is accompanied by a little pilot fish, the odds are 2 to 1, or the probability 2/3, that the next shark will be so accompanied.
When an event has happened a very great number of times, its happening once again approaches nearly to certainty. If we suppose the sun to have risen one thousand million times, the probability that it will rise again, on the ground of this knowledge merely, is 1,000,000,000 + 1/1,000,000,000 + 1 + 1. But then the probability that it will continue to rise for as long a period in the future is only 1,000,000,000 + 1/2,000,000,000 + 1, or almost exactly 1/2. The probability that it will continue so rising a thousand times as long is only about 1/1001. The lesson which we may draw from these figures is quite that which we should adopt on other grounds, namely, that experience never affords certain knowledge, and that it is exceedingly improbable that events will always happen as we observe them. Inferences pushed far beyond their data soon lose any considerable probability. De Morgan has said,[170] “No finite experience whatsoever can justify us in saying that the future shall coincide with the past in all time to come, or that there is any probability for such a conclusion.” On the other hand, we gain the assurance that experience sufficiently extended and prolonged will give us the knowledge of future events with an unlimited degree of probability, provided indeed that those events are not subject to arbitrary interference.
It must be clearly understood that these probabilities are only such as arise from the mere happening of the events, irrespective of any knowledge derived from other sources concerning those events or the general laws of nature. All our knowledge of nature is indeed founded in like manner upon observation, and is therefore only probable. The law of gravitation itself is only probably true. But when a number of different facts, observed under the most diverse circumstances, are found to be harmonized under a supposed law of nature, the probability of the law approximates closely to certainty. Each science rests upon so many observed facts, and derives so much support from analogies or connections with other sciences, that there are comparatively few cases where our judgment of the probability of an event depends entirely upon a few antecedent events, disconnected from the general body of physical science.
Events, again, may often exhibit a regularity of succession or preponderance of character, which the simple formula will not take into account. For instance, the majority of the elements recently discovered are metals, so that the probability of the next discovery being that of a metal, is doubtless greater than we calculated (p. [258]). At the more distant parts of the planetary system, there are symptoms of disturbance which would prevent our placing much reliance on any inference from the prevailing order of the known planets to those undiscovered ones which may possibly exist at great distances. These and all like complications in no way invalidate the theoretic truth of the formulas, but render their sound application much more difficult.
Erroneous objections have been raised to the theory of probability, on the ground that we ought not to trust to our à priori conceptions of what is likely to happen, but should always endeavour to obtain precise experimental data to guide us.[171] This course, however, is perfectly in accordance with the theory, which is our best and only guide, whatever data we possess. We ought to be always applying the inverse method of probabilities so as to take into account all additional information. When we throw up a coin for the first time, we are probably quite ignorant whether it tends more to fall head or tail upwards, and we must therefore assume the probability of each event as 1/2. But if it shows head in the first throw, we now have very slight experimental evidence in favour of a tendency to show head. The chance of two heads is now slightly greater than 1/4, which it appeared to be at first,[172] and as we go on throwing the coin time after time, the probability of head appearing next time constantly varies in a slight degree according to the character of our previous experience. As Laplace remarks, we ought always to have regard to such considerations in common life. Events when closely scrutinized will hardly ever prove to be quite independent, and the slightest preponderance one way or the other is some evidence of connection, and in the absence of better evidence should be taken into account.