826 



SCIENCE. 



[N. S. Vol. II. No. 51. 



one tosses a coin, it is, in general, impossi- 

 ble to know in advance on which face 

 it will rest. That its behavior in this respect 

 will be governed by the operation of forces 

 and conditions, just as certain and just as 

 definitelj' compelling a given result as is 

 the behavior of the sun and moon in the 

 matter of an eclipse, will not be denied. 

 If in any particular trial we knew all of the 

 forces and conditions which influenced the 

 result we should find that they were never 

 equally balanced between the two possible 

 events, but always predominated in favor 

 of that which actually happened. A com- 

 plete knowledge of antecedent causes would 

 reveal the fact that each of these (to us at 

 present unknown) forces and conditions is 

 subject to other secondary influences which 

 continually change its resultant efiect from 

 one side to the other, and so on , in lower de- 

 gree, to the end that in a very large mnnber of 

 trials the ratio of the number of times the 

 two possible events have occurred becomes 

 very nearly one, to which, indeed, it ap- 

 proximates continuously as the number of 

 trials increases. Note the use of the word 

 ratio in this statement. In a very large 

 number of trials in tossing a coin the number 

 of heads may be always in excess of the tails 

 and by a continually increasing amount, 

 and yet the ratio of the two may be con- 

 tinually tending towards equalitj'. It is 

 important to call attention to the depend- 

 ence of the Theory of Chances upon experi- 

 ence and experiment. It is not rigorously 

 true, as is often stated in writing about 

 probabilities, that if a coin is tossed in the 

 air " it is' as likelj- to fall upon one face as 

 another." Such a condition necessitates an 

 absolutely equal division of all forces and 

 conditions between the two possible events, 

 and it is logical to conclude that neither 

 would happen. A more nearly correct 

 statement would be that we are quite ig- 

 norant of any cause tending to one result 

 rather thaa to another. As there are, ap- 



parently, but two possible results we may 

 put the a jmori probability of each at one- 

 half. This conclusion is, however, of little 

 value until experience has proved that in 

 the case under consideration the controlling 

 forces and conditions are so evenly dis- 

 tributed and nicely adjusted that the balance 

 is easily thrown from one side to the other. 

 If experience shows that in a certain series 

 of trials there is a marked tendency for a 

 coin to fall on one face rather than the 

 other we are led at once to suspect that 

 there is something in the manner of tossing, 

 or in the nature of the coin itself, or in some 

 other less easily understood condition, which 

 has caused this tendency, and we know that 

 our numerical expression for the probability 

 cannot be correct. Expei'ience, therefore, 

 is essential to any useful apphcation of this 

 doctrine, and experience is valuable only 

 when it is large. The numerical evalua- 

 tion of a probabilitj' must be, at least, one 

 which is not contradictory to experience 

 and, whenever possible, it must be one sup- 

 ported and verified by experiment. 



The above general remarks on the Doc- 

 trine of Chances (with apologies to the many 

 who ai'e quite familiar with the subject) are 

 submitted with a desire to aid in clearing 

 away some of the difficulties which many 

 people encoiinter in trying to understand 

 the usefulness of this most interesting 

 branch of apjjlied mathematics. In scien- 

 tific investigation whenever our knowledge 

 is so nearlj' complete and our mental vision 

 so far reaching that we can trace the prog- 

 ress of the phenomenon under consideration, 

 or of each of its elements from beginning to 

 end, we do not need its aid. In the thous- 

 ands of instances, however, in which pri- 

 mary causes are so obscure and so numerous 

 that we can only know them by their in- 

 tegrated effects, its assistance has proved to 

 be of incalculable value. 



The object of the present article is to re- 

 mind the reader that whenever the number 



