Example 10.It is known that 30%of a certain company's washing machines require service while under warranty,whereas only 10%of its dryers need such service.If someone purchases both a washer and a dryer made by this company,what is the probability that both machines need warranty service? Let A denote the event that the washer needs service while under warranty, Let B defined analogously for the dryer. Then P(A=0.3,P(B)=0.1. Assuming that the two machines function independently of one another, the desired probability is PA∩B)=P(A)×P(B)=0.3x0.1=0.03 The probability that neither machine needs service is P(nB)=P)·PB)=(0.70.9)=0.63 Example 11 Asystem consists of four components,as illustrated in FigThe entire each subsystem are connected in series,a subsystem will work only if both its components work.If components work or fail independently of one another and if Letting (i=1.2.3.,4)be the event that the ith component works,the 4's are mutually independent. The event that the 1-2 subsystem works in A,and similarly,A denotes the event that the 3-4 subsystem works. The event that the entire system works is ()()so P(4n4)U(4nA】=P(4n4)+P(4nA)-P(4n4)n(AnA】 =P(A)P(A)+P(A)-P(A)-P(A)-P(A)-P(A)-P(A) =(0.90.9)+(0.90.9)-(0.90.90.90.9) =0.9636 Example 10. It is known that 30% of a certain company’s washing machines require service while under warranty, whereas only 10% of its dryers need such service. If someone purchases both a washer and a dryer made by this company, what is the probability that both machines need warranty service? Let A denote the event that the washer needs service while under warranty, Let B defined analogously for the dryer. Then P(A)=0.3, P(B)=0.1. Assuming that the two machines function independently of one another, the desired probability is P(AB)  P(A)P(B)  0.30.1 0.03 The probability that neither machine needs service is P(A'B')  P(A')  P(B')  (0.7)(0.9)  0.63. Example 11. A system consists of four components, as illustrated in Fig .The entire system will work if either the 1-2 subsystem works or if the 3-4 subsystem works (since the two subsystems are connected in parallel). Since the two components in each subsystem are connected in series, a subsystem will work only if both its components work. If components work or fail independently of one another and if each works with probability 0.9, what is the probability that the entire system will work (the system reliability coefficient)? Letting Ai (i=1,2,3,4) be the event that the ith component works, the Ai ’s are mutually independent. The event that the 1-2 subsystem works in A1  A2 , and similarly, A3  A4 denotes the event that the 3-4 subsystem works. The event that the entire system works is ( ) ( ) A1  A2  A3  A4 , so 0.9636 (0.9)(0.9) (0.9)(0.9) (0.9)(0.9)(0.9)(0.9) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [( ) ( )] ( ) ( ) [( ) ( )] 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4                        P A P A P A P A P A P A P A P A P A A A A P A A P A A P A A A A