76 
On Errors of Random Sampling 
A few arithmetical results may now he given. 
Let n = 100 and m = 50. 
Area* of "Normal" 
From Series ('2) Curve, with S. D. \^mpq 
p='4, q = -6 
Chance of 20—22 Successes "255t -271 
Chance of 5—7 Successes -3193 -4739 
0—2 „ 1311 -1145 
^=•01, q = -Q9 
Chance of 0—2 Successes '8938 -9202 
„ 3—5 „ -1007 -0022 
We see how the liability to error increases with p ~q. 
An interesting special case may be discussed here which emphasizes the 
importance of the problem indicated. 
Suppose the first sample has given all successes or all failures, so that p or 
q = 0, how are we to measure its reliability ? 
Many unsophisticated users of formulae must have been puzzled by this case, 
since, construing the formulae au pied de la lettre, it would appear that after 
n successes in n trials, we ought to get m successes in m with a probable error 
o/O! 
The paradox vanishes if we consider (2). Put in it n — p and we have 
m!(» + l). ( 1 + >1+1 (»+!)("+ 2) + et I (7) . 
(n + m + 1) ! { 1 ! 2 ! 
From this we see that the ratio of the (m + l)th term to the whole sum (i.e. the 
n + 1 
chance of m successes in m trials) is — ; from which we conclude : 
n + m + 1 
(a) Only when n is very large as compared with m does the chance of 
obtaining 100 3 / 0 successes in m trials approach unity. 
(6) In particular if n = m and both are large, the chance is about "5. 
For instance if we have had 100 °/ 0 successes in 200 trials the chance of getting 
the same proportion in a subsequent 50 is about 4 to 1. If, on the other hand, 
n = 50 and m = 200 it is 1 to 4. 
In view of what follows it may be worth noticing that a closed expression for 
the sum of any number of terms of (7) can readily be given. 
* Taking for area corresponding to x successes, the area between the ordinates x - - 5 and x + - 5. 
t Approximate only, obtained by using Stirling's theorem in the expression 
m ! (p + r-1) ! (q + m-r + 1) ! 
(m-r + 1) ! p ! (q + m) ! ! ' 
to find the ?'th term of series (2). 
