ON THE AVERAGE RESULT OP OBSERVATIONS. XXV 



equal numbers, out of which, in a large number of 

 drawings, both sorts will come in nearly equal propor- 

 tions, 



But in this, as in every other question of proba- 

 bilities, any additional knowledge of the circumstances 

 which may happen, or have happened, changes the 

 problem, and is equivalent to an extension or limitation 

 of its conditions. When the observations have been 

 made, the position of the observer is altered, since though 

 the law of facility of error be not determined, yet more 

 probability is given to some laws than to others, by 

 inspection of the observations themselves. For instance; 

 if the observations give results of very little discordance, 

 it is immediately obvious that a law of facility which 

 makes the probability of large errors very small, is 

 more likely to have been that which actually existed 

 than one of a different character. The problem now 

 presents an analogy with that of an urn, from which 

 drawings have been made and registered, so that the 

 contents of the urn are to be guessed at from the 

 drawings. 



In the first problem, and supposing that a method of 

 combining the observations is to be chosen before obser- 

 vations made, it is demonstrable that the average of the 

 results is more likely to be true than any other magni- 

 tude. And the same conclusion seems probable in the 

 second case, since unassisted common sense would never 

 draw any distinction between the two problems. But the 

 results of calculation applied to the development of the 

 distinction just drawn, show that the average of obser- 

 vations made is not necessarily the most probable re- 

 sult, nor can be such for more than two observations, 

 unless one particular law of facility of error be sup- 

 posed, which law is the standard law described in 

 Chapter VII. But it is also shown, as mentioned in 

 page 142, that the results of any law of facility, when 

 applied to tolerably large numbers of observations, are 

 nearly identical with those of some variety or other of 

 the standard law; so that, practically. f ^ ~*w>v<* nf 



