692 APPENDIX 



SECTION V APPLICATION OF THE THEORY OF 

 PROBABILITY 



The reader should understand thoroughly that what is commonly known 

 as a " law of nature " is a generalization based upon experience, and that such 

 a law cannot be proved in the strictly logical sense, but. only in the sense of 

 establishing a high degree of probability in its favor. To illustrate, we may 

 take one of the best-established facts of physical science, namely, that all free 

 bodies are attracted by the earth. The evidence for this statement consists 

 in the fact that the thousands and even millions of bodies which have been 

 observed have, without exception, followed this rule. This has established 

 a very high degree of probability. It is altogether conceivable, however, 

 that some body exists which would be repelled by the earth. Although 

 experience has established an overwhelming probability against such an 

 occurrence, we must not overlook the fact that experience has proved the 

 statement only in the sense that it has established a high degree of proba- 

 bility. It has done this and can do nothing more than this. It is doubtful 

 whether any person living has seen a hundred pennies tossed at random, 

 all of which came heads up, and still it is possible that this might happen. 

 Even if no one has seen them fall with all heads up, we are clearly not 

 justified in concluding that there will always be some heads up and some 

 tails up. If a thousand pennies be tossed at random, the probability that 

 they will all fall heads up is so small that we may safely say, if the whole 

 human race were to devote a generation to the tossing of pennies, a thousand 

 at a time, there would still be a very small probability that any one would 

 toss all heads. All this goes to show that certain possible events have such 

 a slight probability that we should not expect them to happen in the lifetime 

 of a given individual. Just so, as time goes on and observations are made in 

 greater numbers, exceptions may be found to any of the so-called " laws of 

 nature." Such exceptions are, however, in many cases exceedingly unlikely. 



It is hoped that the foregoing paves the way for the statement that, 

 while in this subject many results may be stated in terms of probabilities, 

 these results do not differ in reliability on that account from those of any 

 other science based on experience. If a thousand pennies be tossed at 

 random, there is nothing more uncertain than that a given penny will 

 be heads, but it is a matter of common experience that the ratio of the 

 number of heads to the total number of pennies tossed is, in general, nearly 

 one half. We may here recall the statement of Section I, that the theory 

 of probability is needed in this subject because we deal with occurrences 

 and characters of such a nature that we wish to make statements in regard 

 to a large number of them taken together. It is a matter of common experi- 

 ence that results, such as averages and ratios obtained from large numbers 

 of cases, are nearly stationary. We find the average stature of a thousand 

 individuals selected at random from a large population, and are much sur- 

 prised if, upon taking another random sample of a thousand from the same 



