We have hitherto considered the theory of probability only in its simple deductive employment, in which it enables us to determine from given conditions the probable character of events happening under those conditions. But as deductive reasoning when inversely applied constitutes the process of induction, so the calculation of probabilities may be inversely applied; from the known character of certain events we may argue backwards to the probability of a certain law or condition governing those events. Having satisfactorily accomplished this work, we may indeed calculate forwards to the probable character of future events happening under the same conditions; but this part of the process is a direct use of deductive reasoning (p. 226).
Now it is highly instructive to find that whether the theory of probability be deductively or inductively applied, the calculation is always performed according to the principles and rules of deduction. The probability that an event has a particular condition entirely depends upon the probability that if the condition existed the event would follow. If we take up a pack of common playing cards, and observe that they are arranged in perfect numerical order, we conclude beyond all reasonable doubt that they have been thus intentionally arranged by some person acquainted with the usual order of sequence. This conclusion is quite irresistible, and rightly so; for there are but two suppositions which we can make as to the reason of the cards being in that particular order:—
- They may have been intentionally arranged by some one who would probably prefer the numerical order.
- They may have fallen into that order by chance, that is, by some series of conditions which, being unknown to us, cannot be known to lead by preference to the particular order in question.
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.