Past Experience on Future Expectation. 367 



Now suppose that a second trial of m = r + s instances be 

 made, then the probability that the given event will occur 

 r times and fail s, is on the a priori chance being between 

 x and x-\-Bx 



m 



= v r --f—c-a-x)% 



\r[s 



and accordingly the total chance C r , whatever x may be, of 

 the event occurring r times in the second series, is 



m \^xi ] + r (l-x)v+ s clx 



L t f Q «P (l-X)Ulr 



This is, with a slight correction, Laplace's extension of 

 Bayes' Theorem. 



A further evaluation o£ C,. is then usually made by ex- 

 pressing the integrals in /3-f unctions,these again in 7-£unctions, 

 then using Sterling's Theorem, and after a series of approxi- 

 mations, not always closely investigated, showing that r can 

 be represented by the ordinates o£ a normal curve of errors. 

 The present investigation is intended to replace this by a 

 more rigid and non- approximative process. 



We have : 



] jn_ T(p+r + l)T{q + s + l)t(n + 2) 



r ~ \r \s V (p + 1) T(q + 1) T(n + m + 2) ' 



Now write : 



p T(g+m + l)r(n+2) 



r(0 + l)r(n + m + 2)' ' * 



(iii.) 



then the chances of the successive values of r, 0, 1, 2 . . . . m 

 are the successive terms in the series : 



j- ,n p + l _ „, ( m + i) ( iJ+ i)( /) + 2) 

 1 ^ll^m" 2! (r/ + m)(ry+m-l) 



w(m-l)(m-2) - Q+1)Q + 2)Q + 3) . 1 



+ " 3! ( ry + m )(7 +m _i)(^ + ^_2)' heVC 'J (n - ; 



This is a simple hypergeometrical series, the momental 

 properties of which have already been studied *. 



* Pearson, " On Certain Properties of the Hypergeometrical Series, 

 and on the Fitting of such Series to Observation Polygons on the Theory 

 of Chance," Phil. Mag. Feb. 1899, pp. 236-246. 



