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20                     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

                               q ˆ P…S 1 [ S 2 [ S 3 [ S 4 [ S 5 †;     …2:22†
           where S i  is the complement of S i  and represents a bad ith component. Clearly,
           P S i ) ˆ 1   p i . . Since events S i , i ˆ  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),

             S 1 [ S 2 [ S 3 [ S 4 [ S 5 ˆ S 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 ;

                    [
                                         ‡
           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
               q ˆ P…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 †;

           and, using statistical independence,

                 q ˆ…1   p 1 †‡ p 1 …1   p 2 †‡ p 1 p 2 …1   p 3 †‡ p 1 p 2 p 3 …1   p 4 †
                                                                        …2:23†
                     ‡ p 1 p 2 p 3 p 4 …1   p 5 †
           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 very useful one. Given two events A
           and B associated with a random experiment, probability P AjB)  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 n AB . Hence, given
           that event B has occurred, the relative frequency of A  is then n AB /n B . Thus we
           have, in the limit as n B  becomes large,


                                      n AB  n AB  n B  P…AB†
                             P…AjB†      ˆ        
                                      n B    n   n    P…B†
           This relationship leads to Definition 2.3.








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