254 THE PRINCIPLES OF SCIENCE. [chap. 



adopt the first, second, or third hypothesis, the proba- 

 bility that the result actually noticed would follow is f|, 

 ^1, and -^f. Now it is certain that one or other of these 

 hypotheses must be the true one, and their absolute 

 probabilities are proportional to the probabilities that the 

 observ^ed events would follow from them (pp. 242, 243). All 

 we have to do, then, in order to obtain the absolute pro- 

 bability of each hypothesis, is to alter these fractions in 

 a uniform ratio, so that their sum shall be unity, the 

 expression of certainty. Now, since 27 + 16 + 3 = 46, 

 this will be effected by dividing each fraction by 46, and 

 multiplying by 64. Thus the probabilities of the first, 

 second, and third hypotheses are respectively — 



27 16 3 



46' 46' 46" 



The inductive part of the problem is 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 pro- 

 bability of drawing a white one is certainly f ; and as the 

 probability of the box being so constituted is |4, the com- 

 pound probability that the hox will be so filled and will 

 give a white ball at the next trial, is 



27 3 81 

 -7 X - or ^ . 

 46 4 1 84 



Again, the probability is \§ that the box contains two 



white and two black, nnd 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 

 -, X - or 4-. 



46 2 1 84 



From the third supposition we get in like manner the 

 probability 



3 I 3 



4 X - or 4- . 



46 4 184 

 Since one and not more than one hypothesis can be true, 



