Finding the minimum cut of an undirected graph
There are plenty of algorithms for finding the min-cut of an undirected graph. Karger's algorithm is a simple yet effective randomized algorithm.

In short, the algorithm works by selecting edges uniformly at random and contracting them with self-loops removed. The process halts when there are two nodes remaining, and the two nodes represent a cut. To increase the probability of success, 
the randomized algorithm is ran several times. While doing the runs, one keeps track of the smallest cut found so far.

See the Wikipedia entry for more details. For perhaps a better introduction, check out the first chapter of Probability and Computing: Randomized Algorithms and Probabilistic Analysis by Michael Mitzenmacher and Eli Upfal.