402 Chance Distribution of Cases of Disease in Houses 
Writing n 1 = m 1 a 1 n/N, n 2 = m 2 a 2 n/N, &c, and dealing separately with each class 
of house as in the original problem, we get, by adding the results for the various 
classes together (taking the approximation p 0 =p 0 ' and writing q 0 for the new 
value of p 0 ), 
n i ih n s 
q 0 = m x e mi + m 2 e m - 2 + m z e m * + . . . 
a^n a 2 n 
N~ N~ 
= mjg + m 2 e + 
aiti a 2 n 
— N~ 
q 1 = n 1 e + n 2 e + 
a.\n a 2 n 
nl - w, n£ " T . 
q-i = — in e H s-: e + — 
It is thus a simple matter to calculate the values of q 0 , q lt q 2 , when we have given 
the number of cases of disease, the total population and the number of houses 
averaging (say) 3, 5, 10, &c, inhabitants. An imaginary example will show that 
the change may quite reasonably be a 33 °/ Q increase on the value of p 2 . 
If in a town with 50,000 inhabitants and 10,000 houses there are 500 cases of 
a certain disease, then we find jy 0 = 9512, ^ = 476, p. 2 =12, and p 3 = 0 to the 
nearest whole number ; and p 0 ' = 9512, p/ — 476, p 2 = 12, and p 3 ' = 0. If we now 
assume 3000 houses to have 2 inhabitants, 5000 to have 5, 1000 to have 7, 800 to 
have 10, and 200 to have 20, then q 0 = 9516, q, = 467, q, = 16, and q 3 = l. 
Returning now to our original example we are unfortunately unable to obtain 
a distribution of houses according to number of inhabitants in the town dealt with 
(viz., Manchester). We, therefore, assume that the town would show a somewhat 
similar distribution to that of Baltimore, U.S.A., for which we were able to obtain 
the necessary figures. The following table shows the values found for q 2 on this 
assumption. While it is probably only approximately accurate it will give an 
idea of the change in value of p 2 which may be reasonably expected. 
Columns in Table. 
(1) Number of people to a house. 
(2) Number of houses out of 10,000 with 1, 2, 3, &c, persons (Baltimore). 
(3) Number of houses with 1, 2, 3, &c, people in Manchester if the distribu- 
tion had been the same as that in Baltimore. 
(4) Number of persons in each group of houses. 
(5) Number of cases of Enteric occurring in each group. 
(6) Calculated number of houses affected twice in each group. 
