IN THE THEORY OF PROBABILITIES. 429 



the first and third become equal when ■ar = i ; that is, when A happens 

 as often as B in ti trials: a result which we might have looked for 

 a priori. It also appears that wlien tjt is less than i, it is more likely 

 that p should exceed -sr than fall short of it; which is in accordance 

 with another result of the theory, namely, that the chance of drawing 



A at the (n + 1)"' trial is -^ -, which is nearer to i than —^ 



Let TT = 1 - K where k is small f not being less than -] . 



Let the limits be « = 1 - x,c and ^. = 1 ; where X is greater than 

 unity, or ^ is negative, and v is positive and infinite. We have also 



1 — (C 1 



M- = M (1 - k) log ^ _ ^^ + UK log - = fiK (\ - I - log X) nearly : 



so that if A happen w (1 - «) times out of n, the presumption that its 

 probability lies between 1 - \k, and 1 is 



,- J e at- ^:r -— {^ = -Vmx . Vx - 1 - log X . 



Next, let 70- = 1- «, where k is small fnot being less than i], and 



let « = 0, J = I ; that is, required the presumption that the less frequent 

 (slightly) of two events is the less probable. Then ,x is infinite and 

 negative, and v is positive and derived from 



v^ = n{\-K) log (] - 2/c) + w (1 + »c) log (1 + 2k) 



= 2«K- nearly. 



The presumption required is then 



4=/" c-'VZ< or 1 + 4- f^-'e-^'dt. 



AUGUSTUS DE MORGAN. 



University College, London, 

 December 30, 1837. 



31 2 



