Fundamentals of Probability and Statistics for Engineers

we expect that the ratio nAB/nA also tends to P(B) as nA becomes large. The
independence assumption then leads to the observation that

This then gives

or, in the limit as n becomes large,

which is the definition of independence introduced above.

Example 2.7.In launching a satellite, theprobability of an unsuccessful
launch is q. What is the probability that two successive launches are unsuccess-
ful? Assuming that satellite launchings are independent events, the answer to
the above question is simply q^2. One can argue that these two events are not
really completely independent, since they are manufactured by using similar
processes and launched by the same launcher. It is thus likely that the failures of
both are attributable to the same source. H owever, we accept this answer as
reasonable because, on the one hand, the independence assumption is accept-
able since there are a great deal of unknowns involved, any of which can be
made accountable for the failure of a launch. On the other hand, the simplicity
of computing the joint probability makes the independence assumption attract-
ive. In physical problems, therefore, the independence assumption is often
made whenever it is considered to be reasonable.

Care should be exercised in extending the concept of independence to more
than two events. In the case of three events, A 1 ,A 2 ,andA 3 , for example, they
are mutually independent if and only if

and

Equation (2.18) is required because pairwise independence does not generally
lead to mutual independence. Consider, for example, three events A 1 ,A 2 ,and
A 3 defined by

18 Fundamentals of Probability and Statistics for Engineers

nAB

nA

P…B†

nB

n

:

nAB

n



nA

n

n

B

n



;

P…AB†ˆP…A†P…B†;

P…AjAk†ˆP…Aj†P…Ak†; j6ˆk; j;kˆ 1 ; 2 ; 3 ; … 2 : 17 †

P…A 1 A 2 A 3 †ˆP…A 1 †P…A 2 †P…A 3 †: … 2 : 18 †

A 1 ˆB 1 [B 2 ; A 2 ˆB 1 [B 3 ; A 3 ˆB 2 [B 3 ;