A fluid flow system connecting points X and Y is illustrated

A fluid flow system connecting points X and Y is illustrated schematically below. Fluid can only flow through each labeled junction when that \"switch\" is closed. Each switch is independent of the others, and the probability that any given switch is open is 80%. What is the probability that fluid flows from X to Y? What is the probability that fluid flows from X to Y, given that switch D is open? Suppose switch E always copies the status of switch A. What is the probability that fluid flow\'s from X to Y?

Solution

a) probability that each switch is open is 0.8

so probability of each switch is close is 0.2

fluid will flow only when all the switch are close

there are 2 ways for fluid to reach from x to y

x-a-c-d-y or x-b-c-e-y

therefore probability will be 0.2*0.2*0.2*0.2*0.2 +  0.2*0.2*0.2*0.2*0.2 = 0.00064

b) if d is open then we will take the case of x-b-c-e-y =  0.2*0.2*0.2*0.2*0.2 = 0.00032

c)if A is open then E is also open so probility will be 0 as fluid will not flow in any case.

if A is CLOSE then E is also close so the probability will be same as in case 1) = 0.00064