M. Greenwood 
73 
or " population " from which the samples come. This problem of drawing from 
a " limited universe" will not be considered in the present memoir; it has been 
discussed in the paper of Pearson last cited*. 
The class of problem to which attention is now directed may be typified as 
follows : — 
Fifteen " control " rats have been inoculated with a constant dose of a standard 
culture of plague bacilli and twelve succumbed in a certain time. Ten similar rats 
have been immunised by a method it is desired to test and five of these died after 
inoculation with a dose of culture similar to that employed upon the "controls." 
What is the probability that the deviation from the rate of mortality obtaining 
among the " controls " is a chance event ? 
Evidently the methods of pp. 69 — 72 cannot be used. To state that the a priori 
chance of dying is "8 is to ignore the fact that the size of the "control" sample 
does not justify us in assuming that its proportional yield approximates at all 
closely to that of the whole population. 
Let us, then, start from first principles, merely assuming (an assumption based 
on or supported by the fairly wide practical experience of civilised humanity) that 
all possible events are, in the absence of any grounds for inference, equally likely 
(Bayes' principle). 
On this assumption, we have, by Bayes' Theorem for the chance P x that the 
true probability of an event, observed to happen p and fail q times in n trials, is 
between x and x + Sx : 
xp (1 - x)i dx 
"as -- ji • 
xP(l-x)idx 
J 0 
A second trial of m being made, the total chance of its yielding r successes 
and m — r failures is: 
\ x? +r (l - x)i+'"- r dx 
m ! 
r ! (m - r) ! " f 1 
f x* (1 -x)*dx 
.(1). 
This is, in modern notation, the result contained in the 7th of Condorcet's 
problems published in his Essai, 1785 "f", but Laplace had, eleven years previously, 
m ! 
given the theorem with the omission of the term — — — (i.e. working on the 
5 r ! (m - r) ! & 
standard model of an urn from which balls are drawn, he assumed the drawings to 
have been made in an assigned order). 
To Pearson \, whose symbols will be used, belongs the credit of emphasizing the 
enormous statistical value of the theorem. The usual method of treating (1) has 
* See Pearson, op. cit. pp. 173 — 5. 
t See Todhunter's History of the Theory of Probability, p. 383, and for a similar result obtained by 
a different process in 1795 by Prevost and Lhuilier, op. cit. p. 453. 
X Karl Pearson, "On the Influence of Past Experience on Future Expectation," Philosophical 
Magazine, 1907, p. 365. 
Biometrika ix 10 
