CAUSE AND EFFECT— PROBABILITY 145 



experience of two tosses both resulting in heads. On the 

 first constitution of the body this would be a certain 

 result, or its probability be represented by i ; on the 

 second constitution the result would be impossible, or the 

 probability would be zero, while on the third constitution 

 — that of the customary coin — the probability of the 

 result would be ^. Experience, then, shows us that one 

 constitution of the coin is impossible, and that another 

 constitution will certainly give the observed result, while 

 the odds against the remaining possible constitution 

 giving it are 3:1. Obviously a double head is a more 

 probable constitution for the coin than head and tail. 

 But in what ratio is this constitution more probable than 

 the other ? This is determined by a principle due to 

 Laplace, which we may state as follows : — 



" If a result might flow from any one of a certain 

 number of different constitutions, all equally probable 

 before experience, then the several probabilities of each 

 constitution after experience being the real constitution, 

 are proportional to the probabilities that the result would 

 flow from each of these constitutions." 



Thus in our case the head-head constitution gives a 

 probability of i that the observed result will arise, while 

 head-tail only gives a probability of \. Hence, on 

 Laplace's principle, the odds are four to one that our 

 coin has a head on both sides. We must be careful to 

 note that this result depends entirely on the assumption 

 that coins may have any constitution whatever ; it ceases 

 to have application when we have once had the experience 

 that coins usually have a head and a tail. But it may 

 be said, ought we not to have had the actual experience 

 that coins may be of any constitution before we can 

 predict that the individual coin which has twice turned up 

 heads is probably a double-headed coin ? Can we assume 

 without such experience that, where we are ignorant, all 

 constitutions are a priori equally probable ? May we for 

 the very reason that we know nothing " distribute our 

 ignorance equally " ? The logic of this proceeding has 

 been called in question by more than one writer, notably 



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