them both less often than once in twelve times, they may belong to the same office, where one acts as a substitute for the other.  Similarly, if in a multitude of throws a die turns six oftener than once in six times, it is not a fair one:  that is, there is a cause favouring the turning of six.  If of 20,000 people 500 see apparitions and 100 have friends murdered, the chance of any man having both experiences is 1/8000; but if each lives on the average 300,000 hours, the chance of both events occurring in the same hour is 1/2400000000.  If the two events occur in the same hour oftener than this, there is more than a chance coincidence.

The more minute a cause of connection or repugnance between events, the longer the series of trials or instances necessary to bring out its influence:  the less a die is loaded, the more casts must be made before it can be shown that a certain side tends to recur oftener than once in six.

(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 x 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.