The latter supposition is by no means absurd, for any one order is as likely as any other when there is no preponderating tendency. But we can readily calculate by the doctrine of permutations the probability that fifty-two objects would fall by chance into any one particular order. Fifty-two objects can be arranged in 52 × 51 × ... × 3 × 2 × 1 or about 8066 × (10)64 possible orders, the number obtained requiring 68 places of figures for its full expression. Hence it is excessively unlikely that anyone should ever meet with a pack of cards arranged in perfect order by accident. If we do meet with a pack so arranged, we inevitably adopt the other supposition, that some person, having reasons for preferring that special order, has thus put them together.
We know that of the immense number of possible orders the numerical order is the most remarkable; it is useful as proving the perfect constitution of the pack, and it is the intentional result of certain games. At any rate, the probability that intention should produce that order is incomparably greater than the probability that chance should produce it; and as a certain pack exists in that order, we rightly prefer the supposition which most probably leads to the observed result.
By a similar mode of reasoning we every day arrive, and validly arrive, at conclusions approximating to certainty. Whenever we observe a perfect resemblance between two objects, as, for instance, two printed pages, two engravings, two coins, two foot-prints, we are warranted in asserting that they proceed from the same type, the same plate, the same pair of dies, or the same boot. And why? Because it is almost impossible that with different types, plates, dies, or boots some apparent distinction of form should not be produced. It is impossible for the hand of the most skilful artist to make two objects alike, so that mechanical repetition is the only probable explanation of exact similarity.
We can often establish with extreme probability that one document is copied from another. Suppose that each document contains 10,000 words, and that the same word is incorrectly spelt in each. There is then a probability of less than 1 in 10,000 that the same mistake should be made in each. If we meet with a second error occurring in each document, the probability is less than 1 in 10,000 × 9999, that two such coincidences should occur by chance, and the numbers grow with extreme rapidity for more numerous coincidences. We cannot make any precise calculations without taking into account the character of the errors committed, concerning the conditions of which we have no accurate means of estimating probabilities. Nevertheless, abundant evidence may thus be obtained as to the derivation of documents from each other. In the examination of many sets of logarithmic tables, six remarkable errors were found to be present in all but two, and it was proved that tables printed at Paris, Berlin, Florence, Avignon, and even in China, besides thirteen sets printed in England between the years 1633 and 1822, were derived directly or indirectly from some common source.[150] With a certain amount of labour, it is possible to establish beyond reasonable doubt the relationship or genealogy of any number of copies of one document, proceeding possibly from parent copies now lost. The relations between the manuscripts of the New Testament have been elaborately investigated in this manner, and the same work has been performed for many classical writings, especially by German scholars.
Principle of the Inverse Method.
The inverse application of the rules of probability entirely depends upon a proposition which may be thus stated, nearly in the words of Laplace.[151] If an event can be produced by any one of a certain number of different causes, all equally probable à priori, the probabilities of the existence of these causes as inferred from the event, are proportional to the probabilities of the event as derived from these causes. In other words, the most probable cause of an event which has happened is that which would most probably lead to the event supposing the cause to exist; but all other possible causes are also to be taken into account with probabilities proportional to the probability that the event would happen if the cause existed. Suppose, to fix our ideas clearly, that E is the event, and C1 C2 C3 are the three only conceivable causes. If C1 exist, the probability is p1 that E would follow; if C2 or C3 exist, the like probabilities are respectively p2 and p3. Then as p1 is to p2, so is the probability of C1 being the actual cause to the probability of C2 being it; and, similarly, as p2 is to p3, so is the probability of C2 being the actual cause to the probability of C3 being it. By a simple mathematical process we arrive at the conclusion that the actual probability of C1 being the cause is
p1/p1 + p2 + p3;
and the similar probabilities of the existence of C2 and C3 are,
p2/p1 + p2 + p3 and p3/p1 + p2 + p3.
The sum of these three fractions amounts to unity, which correctly expresses the certainty that one cause or other must be in operation.