Karl Pearson 
For a given position between x and .^'4■ Zx, Bayes takes the total probability to be 
^x /xy A ,ry 
a \a/ \ a) 
It is against this first stage that the objection is raised that the chance of placing 
the first stroke is taken as Krjii, and of succeeding strokes xja or {a — x)/a. All 
this is on the assumption that the first stroke a priori is equally likely to be 
anywhere in the line of length a, or freed from the analogy of strokes, the a priori 
chance is taken to have any value between 0 and 1 with equal likelihood. But 
there is really no necessity for this limitation. Suppose the frequency curve for 
<f) (x) 
strokes along the line to be given by any continuous function of x, say y — . 
Then the chance of a stroke occurring between x and x + hx will be 0 {x) Bx/a 
instead of 8xja. Further, the chance Px of a stroke afterwards occurring between 
0 and X will be 
P,= r(f>{x)dx/a, 
J (I 
and between x and a Q_i- = 1 — P^ = I </> (x) dxja. 
J X 
Clearly Po= 0 and Pf, = 1. Thus the probability of the combined event, since 
BP., = (f,{x)8x/a, 
will be hP.Pj'Q, 
(p-¥ q)l 
plq 
Proceeding as in Bayes' or Laplace's manner, we have for the probability that the 
unknown original probability lies between Po and P,. (i.e. x between b and c) 
dP,p,^i-p,y' 
r 
'dp,p,^{i-p,)^ 
0 
Or, again, for the whole chance that there will be in a further m. trials, r successes 
and s failures, 
dP,P,nl-Px)'' 
J (I 
Now it is clear that the above integrals will take the same values 
B(p-\-r + l,q + s + l} and B(p + l,q + l) 
whether we replace Px by x or not. But if we replace P^ by x, we have exactly 
the formula reached by Laplace on the basis of " the equal distribution of 
ignorance," to use Boole's phrase. Thus it would appear that the fundamental 
formula of Laplace*, i.e. 
^ Bjp + r + l, q + s + 1) 
''^ B{p + l,q + l)B{r + l,s + l)' 
* lucludinR the H {r + 1, .s + 1) term omitted by Laplace. 
