136 ESSAY ON PROBABILITIES. 



combined with such a negative error as chance may 

 yield. That is, he must suppose all the negative 

 errors altered by the introduction of such an additional 

 positive error, and each of the positive errors increased 

 or reduced to S -5- 2s. And the same if he would 

 compound for negative errors only : while, to compound 

 for both, he must suppose every observation affected 

 both by a positive error of S -f- 2s and by a negative 

 error of the same amount. This latter case supposes 

 every observation to be correct, which is the result in 

 the long run. The use of this consideration is, to keep 

 before the mind the average effect of positive error, not 

 upon those observations which have positive error, but 

 upon all the observations; and the same for negative 

 errors. Suppose I have an instrument which makes 

 positive and negative errors in equal numbers and to 

 equal amounts, in the long run ; and suppose that it is 

 in my power totally to destroy positive error, leaving 

 the chance of negative error as before. What is the 

 effect upon the whole system of errors, positive and 

 negative, one with another ? What must I do with all 

 the errors to reproduce the same amount of absolute 

 error as before ? I must affect every observation with 

 one half of the average amount of positive error, or one 

 half the magnitude which the positive errors have, one 

 with another. Or look at it in this way ; if I have to 

 pay a shilling for every unit of positive error, for how 

 much should another person take the risk off my hands, 

 that is, for how much per observation, whether its 

 error be positive or negative. If the average positive 

 error were 1, I should have in the long run to pay at 

 the rate of one shilling for every two observations, against 

 which I should insure at the rate of sixpence per 

 observation. 



The mean risk of positive error, then, is the average 

 positive error, when the errors of this kind are equally 

 distributed over all the observations : and the same for 

 negative error. When positive and negative error are 



