Question

The figure below (Figure 2) shows the pipe layout in a lawn sprinkler system. Sprayers are located atevery intersection of the pipe and also at the corners of the system. Thus there are 16sprayers. Two things can happen to a sprayer: it can break off or it can get clogged.If any one of the sprayers breaks off, a great stream of water will pour forth and thesystem will have failed. If one of the sprayers clogs, it will fail to water the lawn inthe immediate vicinity; however, adjacent sprayers can reach the affected areas. Thesystem is judged to be failed if any two adjacent sprayers are both failed. For one sprayer, let the probability of breakage be 0.01 and the probability of clogging be 0.05. Find the reliability of the system (the probability that it does not fail in either mode

Answer 1

Let A denotes sprayers breaks off

P(A) =0.01

P(A^c)=1-P(A)

=1-0.01

=0.99

B denotes sprayers clogs

P(B) =0.05

P(B^c) =1- P(B)

=1-0.05

=0.95

The reliability of the system

=probability that it does not fail in either mode

=P(A^c $\cap$ B^c)

=P(A^c)* P(B^c) , A and B are independent event

=0.99 * 0.95

=0.9405