CALCULATION OF CHANCES. 79 



the two quantities, the product being the only function 

 of the two which obeys that particular law of varia- 

 tion. Therefore, the probability that M was produced 

 by either cause, is as the antecedent probability of 

 the cause, multiplied by the probability that if it 

 existed it would produce M. Which was to be demon- 

 strated. 



Or we may prove the third case as we proved the 

 first and second. Let A be twice as frequent as B ; 

 and let them also be unequally likely, when they exist, 

 to produce M : let A produce it twice in four times, 

 B thrice in four times. The antecedent probability of 

 A is to that of B as 2 to 1 ; the probabilities of their 

 producing M are as 2 to 3 ; the product of these ratios 

 is the ratio of 4 to 3, which therefore, if the theorem 

 be true, will be the ratio of the probabilities that A or 

 B was the producing cause in the given instance. And 

 such will that ratio really be. For, since A is twice 

 as frequent as B, out of twelve cases in which one or 

 other exists, A exists in 8 and B in 4. But of its 

 eight cases, A, by the supposition, produces M in only 

 4, while B of its four cases produces M in 3. M, 

 therefore, is only produced at all in seven of the 

 twelve cases ; but in four of these it is produced by 

 A, in three by B ; hence, the probabilities of its 

 being produced by A and by B are as 4 to 3, and are 

 expressed by the fractions f and y . Which was to be 

 demonstrated. 



It is here necessary to point out another serious 

 oversight in Laplace's theory. When he first intro- 

 duces the foregoing theorem, he characterises it cor- 

 rectly, as the principle for determining to which of 

 several causes we are to attribute a known fact. But 

 after having conceived the principle thus accurately, 



