6i 4 THE POPULAR SCIENCE MONTHLY. 



New York ; and, second, the average number of teeth in the head of 

 a New York child then the product of these two numbers would 

 give the average number of children's teeth in a New York family. 

 But this mode of reckoning will only apply in general under two re- 

 strictions. In the tirst place, it would not be true if the same child 

 could belong to different families, for in that case those children who 

 belonged to several different families might have an exceptionally 

 large or small number of teeth, which would affect the average num- 

 ber of children's teeth in a family more than it would affect the aver- 

 age number of teeth in a child's head. In the second place, the rule 

 would not be true if different children could share the same teeth, the 

 average number of children's teeth being in that case evidently some- 

 thing different from the average number of teeth belonging to a child. 



In order to apply this rule to probabilities, we must proceed as fol- 

 lows : Suppose that we have given the probability that the conclusion 

 B follows from the premise A, B and A representing as usual certain 

 classes of propositions. Suppose that we also knew the probability of 

 an inference in which B should be the premise, and a proposition of a 

 third kind, C, the conclusion. Here, then, we have the materials for 

 the application of this rule. We have, first, the relative number of B's 

 per A. We next should have the relative number of C's per B fol- 

 lowing from A. But the classes of propositions being so selected that 

 the probability of C following from any B in general is just the same 

 as the probability of C's following from one of those B's which is de- 

 ducible from an A, the two probabilities may be multiplied together, 

 in order to give the probability of C following from A. The same 

 restrictions exist as before. It might happen that the probability that 

 B follows from A was affected by certain propositions of the class B 

 following from several different propositions of the class A. But, 

 practically speaking, all these restrictions are of very little conse- 

 quence, and it is usually recognized as a principle universally true 

 that the probability that, if A is true, B is, multiplied by the proba- 

 bility that, if B is true, C is, gives the probability that, if A is true, 

 Cis. 



There is a rule supplementary to this, of which great use is made. 

 It is not universally valid, and the greatest caution has to be exercised 

 in making use of it a double care, first, never to use it when it will 

 involve serious error ; and, second, never to fail to take advantage of it 

 in cases in which it can be employed. This rule depends upon the fact 

 that in very many cases the probability that C is true if B is, is substan- 

 tially the same as the probability that C is true if A is. Suppose, for 

 example, we have the average number of males among the children 

 born in New York ; suppose that we also have the average number of 

 children born in the winter months among those born in New York. 

 Now, we may assume without doubt, at least as a closely approxi- 

 mate proposition (and no very nice calculation would be in place in 



