Fundamentals of Probability and Statistics for Engineers

An expression for q may also be obtained by noting that the system fails if
any one or more of the five components fail, or

whereSi is the complement of Si and represents a bad ith component. Clearly,

. Since events 1,... , 5, are not mutually exclusive, the
calculation of q with use of Equation (2.22) requires the use of Equation (2.15).
Another approach is to write the unions in Equation (2.22) in terms of unions of
mutually exclusive events so that Axiom 3 (Section 2.2.1) can be directly utilized.
The result is, upon applying the second relation in Equations (2.10),

where the ‘ ’ signs are replaced by ‘ ’ signs on the right-hand side to stress the
fact that they are mutually exclusive events. Axiom 3 then leads to

and, using statistical independence,

Some simple algebra will show that this result reduces to Equation (2.21).

Let us mention here that probability p is called the reliability of the system in
systems engineering.

2.4 Conditional Probability

The concept of conditional probability is a ve ry useful one. Given two eve nts A
and B associated with a random experiment, probability is defined as
the conditional probability of A, given that B has occurred. Intuitively, this
probability can be interpreted by means of relative frequencies described in
Example 2.6, except that events A and B are no longer assumed to be independ-
ent. The number of outcomes where both A and B occur is nAB. Hence, gi ven
that event B has occurred, the relative frequency of A is then nAB/nB. Thus we
have, in the limit as nB becomes large,

This relationship leads to Definition 2.3.

20 Fundamentals of Probability and Statistics for Engineers

qˆP…S 1 [S 2 [S 3 [S 4 [S 5 †; … 2 : 22 †

P Si)ˆ 1 pi. Si,iˆ

S 1 [S 2 [S 3 [S 4 [S 5 ˆS 1 ‡S 1 S 2 ‡S 1 S 2 S 3 ‡S 1 S 2 S 3 S 4 ‡S 1 S 2 S 3 S 4 S 5 ;

[ ‡

qˆP…S 1 †‡P…S 1 S 2 †‡P…S 1 S 2 S 3 †‡P…S 1 S 2 S 3 S 4 †‡P…S 1 S 2 S 3 S 4 S 5 †;

qˆ… 1 p 1 †‡p 1 … 1 p 2 †‡p 1 p 2 … 1 p 3 †‡p 1 p 2 p 3 … 1 p 4 †

‡p 1 p 2 p 3 p 4 … 1 p 5 †

… 2 : 23 †

P AjB)

P…AjB†

nAB

nB

ˆ

nAB

n

nB

n



P…AB†

P…B†