294 THE PRINCIPLES OF SCIENCE. 



multiplying by 64. Thus the probabilities of the first, 

 second, and third hypotheses are respectively 



46 46 46 



The inductive part of the problem is now completed, since 

 we have found that the urn most likely contains three 

 white and one black ball, and have assigned the exact 

 probability of each possible supposition. But we are now 

 in a position to resume deductive reasoning, and infer the 

 probability that the next drawing will yield, say a white 

 ball. For if the box contains three white and one black 

 ball, the probability of drawing a white one is certainly f ; 

 and as the probability of the box being so constituted is 

 f|, the compound probability that the box will be so filled 

 and will give a white ball at the next trial, is 



27 3 81 

 _JL x 2 or _ 



46 4 104 



Again, the probability is ~ that the box contains two 

 white and two black, and under those conditions the 

 probability is | that a white ball will appear ; hence the 

 probability that a white ball will appear in consequence 

 of that condition, is 



16 i 32 



~r x or - 1 ^ 



46 2 184 



From the third supposition we get in like manner the 



probability 



i 



jL x _ or -^ . 

 46 4 184 



Now since one and not more than one hypothesis can be 

 true, we may add together these separate probabilities, 

 and we find that 



81 i 32 3 116 

 184 184 784 184 



is the complete probability that a white ball will be next 

 drawn under the conditions and data supposed. 



