Or we may prove the third case as we proved the first and second. Let A be twice as frequent as B; and let them also be unequally likely, when they exist, to produce M: let A produce it twice in four times, B thrice in four times. The antecedent probability of A is to that of B as 2 to 1; the probabilities of their producing M are as 2 to 3; the product of these ratios is the ratio of 4 to 3: and this will be the ratio of the probabilities that A or B was the producing cause in the given instance. For, since A is twice as frequent as B, out of twelve cases in which one or other exists, A exists in 8 and B in 4. But of its eight cases, A, by the supposition, produces M in only 4, while B of its four cases produces M in 3. M, therefore, is only produced at all in seven of the twelve cases; but in four of these it is produced by A, in three by B; hence, the probabilities of its being produced by A and by B are as 4 to 3, and are expressed by the fractions 4/7 and 3/7. Which was to be demonstrated. § 6. It remains to examine the bearing of the doctrine of chances on the peculiar problem which occupied us in the preceding chapter, namely, how to distinguish coincidences which are casual from those which are the result of law; from those in which the facts which accompany or follow one another are somehow connected through causation. The doctrine of chances affords means by which, if we knew the _average_ number of coincidences to be looked for between two phenomena connected only casually, we could determine how often any given deviation from that average will occur by chance. If the probability of any casual coincidence, considered in itself, be _1/m_, the probability that the same coincidence will be repeated _n_ times in succession is _1/m^n_. For example, in one throw of a die the probability of ace being 1/6; the probability of throwing ace twice in succession will be 1 divided by the square of 6, or 1/36. For ace is thrown at the first throw once in six, or six in thirty-six times, and of those six, the die being cast again, ace will be thrown but once; being altogether once in thirty-six times. The chance of the same cast three times successively is, by a similar reasoning, 1/6^3 or 1/216: that is, the event will happen, on a large average, only once in two hundred and sixteen throws. We have thus a rule by which to estimate the probability that any given series of coincidences arises from chance; provided we can measure correctly the probability of a single coincidence. If we can obtain an equally precise expression for the probability that the same series of coincidences arises from causation, we should only have to compare the numbers. This however, can rarely be done. Let us see what degree of approximation can practically be made to the necessary precision.