Question

Question 1 (20 points: Events, counting, and properties f probabniny Consider the network shown below. There are two kinds of links in the network. Each link of kind o-p +0 fails with probability p and that of kind O 4+0 fails with probability q. Each link is assumed to fail independently of the other. We say that a path is successful if no link in the path fails. For example, the path S-B-T succeeds if none of the links S - B and B-Tfail. (a) i. [2 points] What is probability that the path S- A is successful? i [I point] What is probability that the path S-B-Tis successful? ii. [1 point] What is probability that the path S- C-Tis successful? (b) [4 points] We say that the network is successful if at least one path from S to is successful. What is probability that the above network is successful? (c) Consider the event E (none of the links connected to S fail) i. [2 points] Calculate P(E) ii. [2 points] Given that the event E occurs, what is the probability of the network being succesful? (none of the links of type ofail (d) Consider the event F i. [2 points] Calculate P(F) ti. [4 points] Given that network has been successful, what is the probability that the event F occured? (e) [2 points] Calculate PE U r)

Answer 1

Dear Student, the question is pretty huge, i have solved almost 60% of the question with the answers and explanation, post one more question to get the remaining answers.

(a)

(i) Probability that the path S-A-T is successful = Probability that path S-A is successful * Probability that path A-T is successful = (1-p)*(1-p) = (1-p)^2 = 1 + p^2 - 2p

(ii) Probability that the path S-B-T is successful = Probability that path S-B is successful * Probability that path B-T is successful = (1-p)*(1-q) = (1-p)^2 = 1 - p - q + pq

(iii) Probability that the path S-C-T is successful = Probability that path S-C is successful * Probability that path C-T is successful = (1-q)*(1-q) = (1-q)^2 = 1 + q^2 - 2q

(b) The network will be successful if one of the networks is active and working.

P(network is successful) = P(SAT is successful) + P(SBT is successful) + P(SCT is succesful) = 3 + (p^2+q^2+pq) - (3p+3q)

(c) (i) None of the links connected to S fail implies all the three links (S-A), (S-B) and (S-C) are working, which implies with the probability of

(1-p) * (1-p) * (1-q)

(ii)The network will be successful with the probability of p+2*q

Note - Post any doubts/queries in comments section.