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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