SUPPLEMENTS—THE LAW OF SMALL CHANCES. 
Ultimately we reach 
2, =| 1 +mg +741 Dee. arti... tn ypu'=1) = Gus 02 z ed 247) 
This expression is of course equivalent to the first m'+ 1 terms of 
the binomial expansion beginning with p™, as the student can 
verify. For instance, if m =n — 2, so that m' = 2, we have 
7d #la-2g +2221 0s - ge] 
= pn-2 2 n-2 n(n—1) n-2,2 
=p" (1=q)* + np (1-9) + =r 7. 2 
=p" + nph-lq %s 2: L) 
Let us now suppose that ¢ is very small, so that > = ratio of 
n 
failures to total trials is also very small. Let us also suppose 
that n is so large that ng =A is finite. Writing ¢ — A and putting 
n 
m=mn—m', (T) becomes 
A\ A= \2 3 Am 
(1-2) (1-3) erat oy . 2 
since Z* and smaller fractions can be neglected. 
n 
But ( 1- Ay is shown in books on algebra to be equal to e-A, 
n 
where e is the base of the natural logarithms, when = is infinite 
and, under similar conditions, 
(1-2-1 
n 
Hence, if n be large and ¢ small, we have 
x2 ing Am 
=e=N 
Pp=e (tere 342+ a =) J Ss 
If we put m'=0, we have the chance that the event succeeds 
every time, and (8) reduces to e-A. Put m'=1, and we get the 
chance that the event shall not fail more than once, e=A(1 + A), 80 
that e-*,X is the chance of exactly one failure, and the terms 
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