According to the view here taken, these are the only fundamental rules for the calculation of chances. An additional one, derived from a different conception of probability, is given in some treatises, which if it be sound might be made the basis of a theory of reasoning. Being, as I believe it is, absolutely absurd, the consideration of it serves to bring us to the true theory; and it is for the sake of this discussion, which must be postponed to the next number, that I have brought the doctrine of chances to the reader’s attention at this early stage of our studies of the logic of science.
FOURTH PAPER
THE PROBABILITY OF INDUCTION[[40]]
I
We have found that every argument derives its force from the general truth of the class of inferences to which it belongs; and that probability is the proportion of arguments carrying truth with them among those of any genus. This is most conveniently expressed in the nomenclature of the medieval logicians. They called the fact expressed by a premise an antecedent, and that which follows from it its consequent; while the leading principle, that every (or almost every) such antecedent is followed by such a consequent, they termed the consequence. Using this language, we may say that probability belongs exclusively to consequences, and the probability of any consequence is the number of times in which antecedent and consequent both occur divided by the number of all the times in which the antecedent occurs. From this definition are deduced the following rules for the addition and multiplication of probabilities:
Rule for the Addition of Probabilities.—Given the separate probabilities of two consequences having the same antecedent and incompatible consequents. Then the sum of these two numbers is the probability of the consequence, that from the same antecedent one or other of those consequents follows.
Rule for the Multiplication of Probabilities.—Given the separate probabilities of the two consequences, “If A then B,” and “If both A and B, then C.” Then the product of these two numbers is the probability of the consequence, “If A, then both B and C.”
Special Rule for the Multiplication of Independent Probabilities.—Given the separate probabilities of two consequences having the same antecedents, “If A, then B,” and “If A, then C.” Suppose that these consequences are such that the probability of the second is equal to the probability of the consequence, “If both A and B, then C.” Then the product of the two given numbers is equal to the probability of the consequence, “If A, then both B and C.”
To show the working of these rules we may examine the probabilities in regard to throwing dice. What is the probability of throwing a six with one die? The antecedent here is the event of throwing a die; the consequent, its turning up a six. As the die has six sides, all of which are turned up with equal frequency, the probability of turning up any one is 1/6. Suppose two dice are thrown, what is the probability of throwing sixes? The probability of either coming up six is obviously the same when both are thrown as when one is thrown—namely, 1/6. The probability that either will come up six when the other does is also the same as that of its coming up six whether the other does or not. The probabilities are, therefore, independent; and, by our rule, the probability that both events will happen together is the product of their several probabilities, or 1/6 x 1/6. What is the probability of throwing deuce-ace? The probability that the first die will turn up ace and the second deuce is the same as the probability that both will turn up sixes—namely, 1/36; the probability that the second will turn up ace and the first deuce is likewise 1/36; these two events—first, ace; second, deuce; and, second, ace; first, deuce—are incompatible. Hence the rule for addition holds, and the probability that either will come up ace and the other deuce is 1/36 + 1/36, or 1/18.
In this way all problems about dice, etc., may be solved. When the number of dice thrown is supposed very large, mathematics (which may be defined as the art of making groups to facilitate numeration) comes to our aid with certain devices to reduce the difficulties.