(3) The rule for calculating the probability of a dependent event is the same as the above; for the concurrence of two independent events is itself dependent upon each of them occurring. My meeting with both A and B in the street is dependent on my walking there and on my meeting one of them. Similarly, if A is sometimes a cause of B (though liable to be frustrated), and B sometimes of C (C and B having no causes independent of B and A respectively), the occurrence of C is dependent on that of B, and that again on the occurrence of A. Hence we may state the rule: If two events are dependent each on another, so that if one occur the second may (or may not), and if the second a third; whilst the third never occurs without the second, nor the second without the first; the probability that if the first occur the third will, is found by multiplying together the fractions expressing the probability that the first is a mark of the second and the second of the third.
Upon this principle the value of hearsay evidence or tradition deteriorates, and generally the cogency of any argument based upon the combination of approximate generalisations dependent on one another or "self-infirmative." If there are two witnesses, A and B, of whom A saw an event, whilst B only heard A relate it (and is therefore dependent on A), what credit is due to B's recital? Suppose the probability of each man's being correct as to what he says he saw, or heard, is 3/4: then (3/4 × 3/4 = 9/16) the probability that B's story is true is a little more than 1/2. For if in 16 attestations A is wrong 4 times, B can only be right in 3/4 of the remainder, or 9 times in 16. Again, if we have the Approximate Generalisations, 'Most attempts to reduce wages are met by strikes,' and 'Most strikes are successful,' and learn, on statistical inquiry, that in every hundred attempts to reduce wages there are 80 strikes, and that 70 p.c. of the strikes are successful, then 56 p.c. of attempts to reduce wages are unsuccessful.
Of course this method of calculation cannot be quantitatively applied if no statistics are obtainable, as in the testimony of witnesses; and even if an average numerical value could be attached to the evidence of a certain class of witnesses, it would be absurd to apply it to the evidence of any particular member of the class without taking account of his education, interest in the case, prejudice, or general capacity. Still, the numerical illustration of the rapid deterioration of hearsay evidence, when less than quite veracious, puts us on our guard against rumour. To retail rumour may be as bad as to invent an original lie.
(4) If an event may coincide with two or more other independent events, the probability that they will together be a sign of it, is found by multiplying together the fractions representing the improbability that each is a sign of it, and subtracting the product from unity.
This is the rule for estimating the cogency of circumstantial evidence and analogical evidence; or, generally, for combining approximate generalisations "self-corroboratively." If, for example, each of two independent circumstances, A and B, indicates a probability of 6 to 1 in favour of a certain event; taking 1 to represent certainty, 1-6/7 is the improbability of the event, notwithstanding each circumstance. Then 1/7 × 1/7 = 1/49, the improbability of both proving it. Therefore the probability of the event is 48 to 1. The matter may be plainer if put thus: A's indication is right 6 times in 7, or 42 in 49; in the remaining 7 times in 49, B's indication will be right 6 times. Therefore, together they will be right 48 times in 49. If each of two witnesses is truthful 6 times in 7, one or the other will be truthful 48 times in 49. But they will not be believed unless they agree; and in the 42 cases of A being right, B will contradict him 6 times; so that they only concur in being right 36 times. In the remaining 7 times in which A is wrong, B will contradict him 6 times, and once they will both be wrong. It does not follow that when both are wrong they will concur; for they may tell very different stories and still contradict one another.
If in an analogical argument there were 8 points of comparison, 5 for and 3 against a certain inference, and the probability raised by each point could be quantified, the total value of the evidence might be estimated by doing similar sums for and against, and subtracting the unfavourable from the favourable total.
When approximate generalisations that have not been precisely quantified combine their evidence, the cogency of the argument increases in the same way, though it cannot be made so definite. If it be true that most poets are irritable, and also that most invalids are irritable, a still greater proportion will be irritable of those who are both invalids and poets.
On the whole, from the discussion of probabilities there emerge four principal cautions as to their use: Not to make a pedantic parade of numerical probability, where the numbers have not been ascertained; Not to trust to our feeling of what is likely, if statistics can be obtained; Not to apply an average probability to special classes or individuals without inquiring whether they correspond to the average type; and Not to trust to the empirical probability of events, if their causes can be discovered and made the basis of reasoning which the empirical probability may be used to verify.
The reader who wishes to pursue this subject further should read a work to which the foregoing chapter is greatly indebted, Dr. Venn's Logic of Chance.