exhibited in Tables of Statistics. 363 



bility of (6 — 1) persons doing so by multiplying it by the factor 



n — b+l ab — b + l 1 



b an—b + 1 , ab — b + l 



an—b + \ 

 / «Z> — Z»Y ab — b \ f-. ab—b \ 



\ an — b)\ an — b — \)"'\ an—n+lJ 

 '/ ab — b + l W ab — b+l \ ab — b + l ' 



\ an — b J\ an—b — 1/ " * an—n+1 



an expression wbicli easily reduces itself to 



n — b + l ab — b + l _ b{a — 2)+n+l 



b an—ab — {n — b) ~ {n — b){a—l)b' 



Except, thei'efore, in the anomalous cases of a = 1 ov n=:b, which 

 would be easily explained, this expi'ession is always greater than 1, 

 In the same way the ratio of the probability of 6 + 1 being 

 the number for the year to that of b being so, is 



n—b ab — b 1 



6 + 1 an — b , ab — b 

 an—b 



/ g6 — 6 — l y ab — b — l \ ^ ab—b-l \ 



\ an — b — lj\ an—b — 2/"'\ an—n + l) 



/ _fl6 — 6_\/ ab—b \ /. _ ab—b \ 



\ an — b—lJ\ an—b — 2j"\ an—n+lJ 



which in the same way reduces itself to 

 n — b ab — b 



6 + 1 an—ab—{n — b — iy 

 the inverse of which is 



{u—l)n-{a-2)b+l 

 "^ {n-b){a-l)b ' 

 which is also positive. 



The number 6 is therefore the most likely one to occur in any 

 year. This is simply, in fact, the result which might easily have 

 been foreseen, namely that the average number is the most pro- 

 bable one to occur as the number of the phsenomena in any one 

 year, and consequently that in a table representing a series of 

 such phainomcna, giving the number occurring in each year, we 

 shall expect to iind the number which represents the average 

 occurring oftcner than any other. 



From tlic above it is easily seen that, though the expression 

 for the probability of any given number occurring is a very com- 



