CHAPTER X.
APPLICATION OF THE CALCULUS OF PROBABILITIES TO THE MORAL SCIENCES.
We have just seen the advantages of the analysis of probabilities in the investigation of the laws of natural phenomena whose causes are unknown or so complicated that their results cannot be submitted to calculus. This is the case of nearly all subjects of the moral sciences. So many unforeseen causes, either hidden or inappreciable, influence human institutions that it is impossible to judge à priori the results. The series of events which time brings about develops these results and indicates the means of remedying those that are harmful. Wise laws have often been made in this regard; but because we had neglected to conserve the motives many have been abrogated as useless, and the fact that vexatious experiences have made the need felt anew ought to have reëstablished them.
It is very important to keep in each branch of the public administration an exact register of the results which the various means used have produced, and which are so many experiences made on a large scale by governments. Let us apply to the political and moral sciences the method founded upon observation and upon calculus, the method which has served us so well in the natural sciences. Let us not offer in the least a useless and often dangerous resistance to the inevitable effects of the progress of knowledge; but let us change only with an extreme circumspection our institutions and the usages to which we have already so long conformed. We should know well by the experience of the past the difficulties which they present; but we are ignorant of the extent of the evils which their change can produce. In this ignorance the theory of probability directs us to avoid all change; especially is it necessary to avoid the sudden changes which in the moral world as well as in the physical world never operate without a great loss of vital force.
Already the calculus of probabilities has been applied with success to several subjects of the moral sciences. I shall present here the principal results.
CHAPTER XI.
CONCERNING THE PROBABILITIES OF TESTIMONIES.
The majority of our opinions being founded on the probability of proofs it is indeed important to submit it to calculus. Things it is true often become impossible by the difficulty of appreciating the veracity of witnesses and by the great number of circumstances which accompany the deeds they attest; but one is able in several cases to resolve the problems which have much analogy with the questions which are proposed and whose solutions may be regarded as suitable approximations to guide and to defend us against the errors and the dangers of false reasoning to which we are exposed. An approximation of this kind, when it is well made, is always preferable to the most specious reasonings. Let us try then to give some general rules for obtaining it.
A single number has been drawn from an urn which contains a thousand of them. A witness to this drawing announces that number 79 is drawn; one asks the probability of drawing this number. Let us suppose that experience has made known that this witness deceives one time in ten, so that the probability of his testimony is 9⁄10. Here the event observed is the witness attesting that number 79 is drawn. This event may result from the two following hypotheses, namely: that the witness utters the truth or that he deceives. Following the principle that has been expounded on the probability of causes drawn from events observed it is necessary first to determine à priori the probability of the event in each hypothesis. In the first, the probability that the witness will announce number 79 is the probability itself of the drawing of this number, that is to say, 1⁄1000. It is necessary to multiply it by the probability 9⁄10 of the veracity of the witness; one will have then 9⁄10000 for the probability of the event observed in this hypothesis. If the witness deceives, number 79 is not drawn, and the probability of this case is 999⁄1000. But to announce the drawing of this number the witness has to choose it among the 999 numbers not drawn; and as he is supposed to have no motive of preference for the ones rather than the others, the probability that he will choose number 79 is 1⁄999; multiplying, then, this probability by the preceding one, we shall have 1⁄1000 for the probability that the witness will announce number 79 in the second hypothesis. It is necessary again to multiply this probability by 1⁄10 of the hypothesis itself, which gives 1⁄10000 for the probability of the event relative to this hypothesis. Now if we form a fraction whose numerator is the probability relative to the first hypothesis, and whose denominator is the sum of the probabilities relative to the two hypotheses, we shall have, by the sixth principle, the probability of the first hypothesis, and this probability will be 9⁄10; that is to say, the veracity itself of the witness. This is likewise the probability of the drawing of number 79. The probability of the falsehood of the witness and of the failure of drawing this number is 1⁄10.
If the witness, wishing to deceive, has some interest in choosing number 79 among the numbers not drawn,—if he judges, for example, that having placed upon this number a considerable stake, the announcement of its drawing will increase his credit, the probability that he will choose this number will no longer be as at first, 1⁄999, it will then be ½, ⅓, etc., according to the interest that he will have in announcing its drawing. Supposing it to be 1⁄9, it will be necessary to multiply by this fraction the probability 999⁄1000 in order to get in the hypothesis of the falsehood the probability of the event observed, which it is necessary still to multiply by 1⁄10, which gives 111⁄10000 for the probability of the event in the second hypothesis. Then the probability of the first hypothesis, or of the drawing of number 79, is reduced by the preceding rule to 9⁄120. It is then very much decreased by the consideration of the interest which the witness may have in announcing the drawing of number 79. In truth this same interest increases the probability 9⁄10 that the witness will speak the truth if number 79 is drawn. But this probability cannot exceed unity or 10⁄10; thus the probability of the drawing of number 79 will not surpass 10⁄121. Common sense tells us that this interest ought to inspire distrust, but calculus appreciates the influence of it.
The probability à priori of the number announced by the witness is unity divided by the number of the numbers in the urn; it is changed by virtue of the proof into the veracity itself of the witness; it may then be decreased by the proof. If, for example, the urn contains only two numbers, which gives ½ for the probability à priori of the drawing of number 1, and if the veracity of a witness who announces it is 4⁄10, this drawing becomes less probable. Indeed it is apparent, since the witness has then more inclination towards a falsehood than towards the truth, that his testimony ought to decrease the probability of the fact attested every time that this probability equals or surpasses ½. But if there are three numbers in the urn the probability à priori of the drawing of number 1 is increased by the affirmation of a witness whose veracity surpasses ⅓.