72 Dr, Young’s remarks on the reduction 
drawn out either at once or in succession. It may then be 
demonstrated, as will appear hereafter, from the number of 
ways in which the respective numbers of each kind of balls 
may happen to be drawn, that there is 1 chance in 12-5- that 
exactly 50 of each kind may be drawn, and an even chance 
that there will not be more than 53 of either, though it still 
remains barely possible that even 100 black balls or 100 
white may be drawn in succession. 
From a similar consideration of the number of combina- 
tions affording a given error, it will be easy to obtain the pro- 
bable error of the mean of a number of observations of any 
kind; beginning first with the simple supposition of the cer- 
tainty of an error of constant magnitude, but equally likely 
to fall on either side of the truth, and then deducing from this 
supposition the result of the more ordinary case of the greater 
probability of small errors than of larger ones. This liability 
to a constant error may be represented, by supposing a counter 
to have two faces, marked o and 2 ; the mean value of an 
infinite number of trials will then obviously be 1, and the 
constant error of each trial will be 1, whether positive or 
negative. 
Now in a combination of n trials with such a counter, if we 
divide the sum of the results by n, the greatest possible error 
of the mean thus found will be 1 ; and the probability of any 
other given error will be expressed by the number of combi- 
nation of the faces of n counters affording that error, divided 
by the whole number of combinations; that is, by the corres- 
ponding coefficient of the binomial ( 1 -j- 1)", divided by 2", 
the sum of the coefficients. The calculation therefore will 
stand thus : 
