NOTE ON THE EXTENT TO WHICH THE DISTRIBUTION 
OF CASES OF DISEASE IN HOUSES IS DETERMINED 
BY THE LAWS OF CHANCE. 
By J McD. TROUP, B.A., M.B., B.C. (Cantab.) and G. D. MAYNARD, F.R.C.S.E. 
There is a problem of some importance in connection with medical investiga- 
tions which we venture to think has not received sufficient attention from the 
mathematical standpoint. Stated briefly it is this : If n cases of a certain disease 
occur in a town with m houses, what are the probabilities that a houses will be 
affected with one case, b houses with two cases, and so on ? In actual practice the 
problem presents itself in this way : a houses are known to have contained one 
case, b houses two cases, c houses three cases, and so on : what evidence is to be 
got of house infection or other disturbing factor from such a distribution, or can it 
be shown that such a distribution may have arisen by chance and be explicable by 
the laws of probability? 
In order to render the problem suitable for mathematical treatment, we 
assume for the present that each house is equally liable to infection, i.e., that each 
house contains the same number of inhabitants, all of whom are equally liable to 
infection, irrespective of age and sex. We also assume that any particular house 
is equally likely to be infected at any time during the period under consideration ; 
in practice this involves the exclusion from the data of known instances of case- 
to-case infection. 
With these assumptions the problem now becomes equivalent to the following. 
If n balls are thrown into m equal compartments which are so arranged that 
it is equally likely that any ball will fall into any compartment, then the pro- 
bability P that they will distribute themselves so that, 
p 0 compartments will contain 0 balls, 
Pi 
Pi 
P* 
1 ball, 
2 balls, 
s 
P = 
and so on, 
mini 
(i)- 
IS 
m n p 0 \ p^. ...p, 
!...(! !)*i(2 !)*»...(« l) Ps ... 
