49 8 ON THE ESTIMATION OF 



Denoting by a the radius of the circular area with- 

 in which his skill would, on the average of an im- 

 mense number of shots, enable him to plant half the 

 total number discharged; and by M the fraction ex- 

 pressing the probability in question, certainty being ex- 

 pressed by i, we shall have 



t* 



,9 



while for H the probability of hitting the same area 

 we have 



//= \ M 



(3.) From these expressions, knowing the value of a, 

 which is the inverse measure of the skill of the shooter 

 (being less the greater that skill), it is easy to calculate 

 his chance of hitting a circle of any given radius in a 

 single shot. And, reversing the question, his skill 

 (measured by the fraction \ ) may be ascertained, by 

 observing what percentage of shots he can plant, on a 

 large average, from a given distance, within a circle of 

 any given radius (r). For that percentage being the 

 numerical expression of his probability of hitting the 

 circle, or the value of H, or i M, M is known, and a 

 will be given by the formula. 



/_ /.Of. 2 / Log. 2 



a=r ' V LOS!. M. - - r - V " Lojr.il. 



Thus, if a marksman be observed to plant 9 per 

 cent of his arrows within a circle of one foot in 

 diameter at the distance of one hundred yards we have 



