Consider the communication network shown in the figure below and suppose that each link can fail with probability p. Assume that failures of different links are independent.
There is a point A, and two line segments originate from point A to the right. The top line segment is labeled 'Link 1' and the bottom line segment is labeled 'Link 3'. Link 1 is then connected on its right to another line segment labeled 'Link 2', and Link 3 is connected on its right to another line segment labeled 'Link 4'. Link 2 and Link 4 are joined on their right ends to a single point, so that Links 1,2,3 and 4 together make a diamond-shape. From the single point at the ends of Links 2 and 4 is another line segment labeled 'Link 5', and its right end is labeled 'B'.
Assume that p=1/3. Find the probability that there exists a path from A to B along which no link has failed. (Give a numerical answer.)
Given that exactly one link in the network has failed, find the probability that there exists a path from A to B along which no link has failed. (Give a numerical answer.)