718 SCIENCE AND HYPOTHESIS 



the path predicted has not always been followed; but, on the whole, 

 much ground has been gained. 



The Calculus of Probabilities 



No doubt the reader will be astonished to find reflections on the 

 calculus of probabilities in such a volume as this. What has that 

 calculus to do with physical science? The questions I shall raise 

 without, however, giving them a solution are naturally raised by 

 the philosopher who is examining the problems of physics. So far is 

 this the case, that in the two preceding chapters I have several times 

 used the words " probability " and " chance." " Predicted facts," as 

 I said above, " can only be probable." However solidly founded a 

 prediction may appear to be, we are never absolutely certain that ex- 

 periment will not prove it false; but the probability is often so great 

 that practically it may be accepted. And a little farther on I added : 

 " See what a part the belief in simplicity plays in our generalizations. 

 We have verified a simple law in a large number of particular cases, 

 and we refuse to admit that this so-often repeated coincidence is a 

 mere effect of chance." Thus, in a multitude of circumstances the 

 physicist is often in the same position as the gambler who reckons 

 up his chances. Every time that he reasons by induction, he more or 

 less consciously requires the calculus of probabilities, and that is why 

 I am obliged to open this chapter parenthetically, and to interrupt 

 our discussion of method in the physical sciences in order to examine 

 a little closer what this calculus is worth, and what dependence we 

 may place upon it. The very name of the calculus of probabilities is 

 a paradox. Probability as opposed to certainty is what one does not 

 know, and how can we calculate the unknown? Yet many eminent 

 scientists have devoted themselves to this calculus, and it cannot be 

 denied that science has drawn therefrom no small advantage. How 

 can we explain this apparent contradiction? Has probability been 

 defined? Can it even be defined? And if it cannot, how can we 

 venture to reason upon it? The definition, it will be said, is very 

 simple. The probability of an event is the ratio of the number of 

 cases favorable to the event to the total number of possible cases. A 

 simple example will show how incomplete this definition is : I 

 throw two dice. What is the probability that one of the two at least 

 turns up a 6 ? Each can turn up in six different ways ; the number of 

 possible cases is 6X6=36. The number of favorable cases is 11; the 

 probability i|. That is the correct solution. But why cannot we 

 just as well proceed as follows ? The points which turn up on the 

 two dice form *-^- =21 different combinations. Among these com- 

 binations, six are favorable; the probability is ^. Now why is the 

 first method of calculating the number of possible cases more legiti- 



