6 
The Fmidmnental Problem of Practical Statistics 
in no way depends on the equal distribution of ignorance. It is sufficient to 
assume any continuous distribution — which may vary from one type of a priori 
probability problem to a second — in order to reach the basis of the fundamental 
problem in practical statistics, i.e. the probability that in a second trial of m events 
following a p, q experience, r will be successful and s fail. 
(4) I now propose to develop the above expression rather more completely than 
Laplace has done. We shall first replace the 5-functions by F-functions. Then we 
shall use Stirling's theorem on the assumption that none of the quantities p, q, r 
or s are small. As we solely need to find the variation of C,. with r, we need not 
trouble about terms not involving r or s. We have 
, ^ 1 r ( p + r + 1) .r {q + s + 1) .r (r + s + 2) 
'■ B{p + l,q + l) T + g + r + s + 2) r(r + l)r (.9 + 1) 
^ e^! T (;» + 2) (p + ry + '- + i(q + sf + ' + - 
~ B {p + 1, g + 1) F {n + m + 2) " ,.'- + 4 / + 4 . ^ ^' 
where ^ 
1 1 \ / 1 1 
12 (p + r) 288 {p + ry V 12 (q + s) (q + s)" 
^ 1 1 \ ^ 1 1 
U +T^ + OQ-«:73+ - 1 + TIT +000:::= + 
V 12/- 288?-- 7 V 12s 288s- 
iTT j_ 1 m . on , 
We now take r = —p + h, s = ~ q — k, 
11^ 11 
and put very approximately 
We may write 
1 1 
+ ■ 
^^QV2(p + r) r2(2 + s) 12r 12s _ 
1 + - ) 
^ j _ e-y(/^F(m + 2) 1 V" ' p) J 
\B{p + l,q + l)V{n + m + 2)) / '' + 4 /^x « + i ^• 
vp) \q/ 
The factor in large curled brackets we will call G ; it does not vary with r. The 
T S 
second fixctor, involving - and - , we will call u, and the remainder is y. We have 
°p q ^ 
log, C, - log, G + log, II + log, 
and we need tlie development only of the last two logarithms. We shall write 
(■H + m)ln = 1 + p, and p\n = P, (//» = Q will give the chances of success and failure 
as estimated on the first sample. We have 
log, n = \p (1 + p) + /, + ^} log (l + p + ^-^ + (1 + p) _ + .1 } l + p-^'j 
9/ 
Ipp + h + -i j log (p + - \pq -h+i} log (^p - ^-^ . 
