Consider the drawing below, of system of five pipes in which each pipe has a valve which is open (allowing fluid to pass through) with probability p, 0 < p < 1, and otherwise closed (blocking the flow of fluid), independently of the other pipes.
Find the probability (as a function of p) that fluid can flow from at least one of the two endpoints on the left side to at least one of the endpoints on the right side.
Under the same assumptions as in problem 5, find the probability that fluid can flow from the left side to the right side in the following set of pipes when p = 1/2.
Note: This can be done by long brute force computations, but there is an insightful discrete mathematics approach. In fact, if we called the first figure a 2-by-1 block and the second a 3-by-2 block, the trick will find the crossing probability for any (n+1)-by-n block when p = 1/2.