Question

Complex device. A device consists of modules B1,. . . , B4. It works if at least one of the assemblies B1 or B2 and if both assemblies B3 and B4 also work. The modules can be independent of each other fail. The failure probabilities for a 10-hour operating period are 0.5 for B1 and B2 and 0.2 for B3 and B4. 1. What is the probability that the device will still work after 10 hours of operation? 2. With what probability do all four components still work after this time?

Answer 1

Let bi denote the event that module Bi will work after 10 hours, i =1,2,3,4.

Probability that the device will still work after 10 hours of operation

= Probability that at least one of the assemblies B1 or B2 and both assemblies B3 and B4 will work

= P(b1$\cup$b2)$\cap$(b3$\cap$b4)

[[ at least one of B1 or B2 will work $\Leftrightarrow$ B1 or B2 or both will work $\Leftrightarrow$ B1$\cup$B2 will work $\Leftrightarrow$ b1$\cup$b2)]]

[[ B3 and B4 will work $\Leftrightarrow$ b3$\cap$b4 ]]

= P[(b1$\cap$b3$\cap$b4)$\cup$(b2$\cap$b3$\cap$b4)] [[by distributive law]]

= P(b1$\cap$b3$\cap$b4) + P(b2$\cap$b3$\cap$b4)

[[ B1 and B2 can fail independently, so B1 and B2 can work independently. The event b1 and b2 are independent which implies b1$\cap$b3$\cap$b4(subset of b1) and b2$\cap$b3$\cap$b4(subset of b2) are also independent ]]

= [P(b1)*P(b3)*P(b4)] + [P(b2)*P(b3)*P(b4)]

[[ All the modules can fail independently. So, all the bi's are independent of each other ]]

= (0.5*0.8*0.8)+(0.5*0.8*0.8)

[[ P(B1 will fail within 10 hours) = 0.5 = P(B2 will fail within 10 hours)

$\Rightarrow$ P(b1) = P(B1 will work after 10 hours) = P(B1 will fail within 10 hours)^c = 1 - P(B1 will fail within 10 hours) =1 - 0.5 =0.5. Similarly, P(b2)=0.5

P(B3 will fail within 10 hours) = 0.2 = P(B4 will fail within 10 hours)

$\Rightarrow$ P(b3) = P(B3 will work after 10 hours) = P(B3 will fail within 10 hours)^c = 1 - P(B3 will fail within 10 hours) =1 - 0.2 =0.8. Similarly, P(b4)=0.8]]

= 0.32 + 0.32 = 0.64

Probability that all four components will work after 10 hours

= P(b1$\cap$b2$\cap$b3$\cap$b4)

= P(b1)*P(b2)*P(b3)*P(b4) [[all modules work independently]]

= 0.5*0.5*0.8*0.8

= 0.16