can be applied with ease in many cases, and which bear all the appearance of truth so far as it can be independently judged by a sound and practiced understanding.
Rules of the Inverse Method.
By the solution of the problem, as described in the last section, we obtain the following series of simple rules.
- To find the probability that an event which has not hitherto been observed to fail will happen once more, divide the number of times the event has been observed increased by one, by the same number increased by two.
If there have been m occasions on which a certain event might have been observed to happen, and it has happened on all those occasions, then the probability that it will happen on the next occasion of the same kind m + 1/m + 2. For instance, we may say that there are nine places in the planetary system where planets might exist obeying Bode’s law of distance, and in every place there is a planet obeying the law more or less exactly, although no reason is known for the coincidence. Hence the probability that the next planet beyond Neptune will conform to the law is 10/11.
- 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,20—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.