Question

Show all your workingS Engineering Uncertainty 1. The water supply for a City C comes from two sources A and B through the water supply network shown in the figure below, with either one of these sources being sufficient in the short-term Source A City C Branch 3 h2 Branc SourceB Events F1, F2 and F3 denote the failure of branch 1, 2, and 3 respectively. It is known that P(F1)0.04, P(F2) - 0.05, and P(F3) 0.02. Assuming the failures of the branches are mutually independent, determine the probability of failure of this water supply system to City C [10%) 2. Consider two river streams flowing past an industrial plant. Let A denote the event that stream a is polluted, and B denote that stream b is polluted. It is determined from historical observations that P(A), P(B) P(A v B 4 a. Determine P(AB) and P(BA) b. Interpret the results c. Determine whether A and B are mutually independent. [10%] [596] [5%]

Answer 1

1) For a given network system the water supply to city fails when both the branches F1 and F2 fails or else branch F3 fails when either of the F1 and F2 branches working.

Given all the events are mutually independent

the probability of failure of this water supply system to city C = P(F1 $\cap$ F2 ) + P(F3) = 0.04*0.05 + 0.02 = 0.022

2)

P(A$\cap$B) = P(A) + P(B) - P(A U B) = 1/5 + 1/4 - 3/8= 3/40

a) P(A|B) = P(A$\cap$B) / P(B) = (3/40)/(1/4) = 3/10

P(B|A) = P(A$\cap$B) / P(A) = (3/40)/(1/5) = 3/8

b) the probability of event A happens given that the event B happens = 3/10

the probability of event B happens given that the event A happens = 3/8

c) P(A$\cap$B) = P(A)*P(B) = (1/5)*(1/4) = 1/20

here P(A$\cap$B) is not equal P(A)*P(B) hence these events are not mutually independent.