624 PROFESSOR BOOLE ON THE COMBINATION 



Here W x =- — — ^ W 2 = 3 — ^ ... (4) 



1 — c 1 1 — c 2 1 — c x 1 — c 2 



If c t = l, we have 



W 1 = l W 2 = 



and r=p y 



This accords with the condition that if either of the observations is believed to 

 be correct, the value which it furnishes for the altitude of the star must be taken 

 as the true one. 



^thly, If c t —c 2 , i.e., if we have no right to give preference to one observation 

 over the other, we have 



r = P -l^ (5) 



the formula of the arithmetical mean. 



Qthly, From the form of W,, W 2 in (4), it is evident that the weights, so to 

 speak, of the observations vary in a higher ratio than that of the simple proba- 

 bilities of correctness of the observations. The practical lesson to be drawn from 

 this is, that we ought to attach a greater weight to good observations, and a 

 smaller to bad ones, than, according to usual modes of consideration, we should 

 be disposed to do. 



The above are the most important observations suggested by the formula 

 to which the last investigation has led. One or two remarks remain to be offered 

 upon the analysis by which it was obtained. 



Although the two forms of investigation which we have exhibited differ, there 

 is nothing inconsistent in the results to which they lead. If we compare corre- 

 sponding formulae in the two, e.g., the values of Prob. w yz, or those of Prob. xyz, 

 we shall find that the one investigation assigns a definite but consistent value to 

 what the other left arbitrary. Either comparison gives 



c _ r—a l c 1 p l — a. 2 c 2 p 2 



We may prove, either by the " conditions of possible experience," or independently, 

 that this value is necessarily a proper positive fraction, and this accords with 

 the interpretation of c as a probability. Art. 23. 



27. But a much more important consideration is the following. It is a plain 

 consequence of the logical theory of probabilities, that the state of expectation 

 which accompanies entire ignorance of an event is properly represented, not by 



the fraction ^ but by the indefinite form ^- And this agrees with a conclusion 



at which Bishop Terrot, on independent, but as I think just grounds, has arrived.* 

 Now this shows, why, if the consideration of the a priori probability of z is, from 



* Transactions of the Royal Society of Edinburgh, vol. xxi. p. 375. 



