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In this work we study two random procedures in cubic graphs with large girth. The first procedure finds a probability distribution on edge-cuts such that each edge is in a randomly chosen cut with probability at least 0.88672. As corollaries, we derive lower bounds for the size of maximum cut in cubic graphs with large girth and in random cubic graphs, and also an upper bound for the fractional cut covering number in cubic graphs with large girth. The second procedure finds a probability distribution on independent sets such that each vertex is in an independent set with probability at least 0.4352. This implies lower bounds for the size of maximum independent set in cubic graphs with large girth and in random cubic graphs, as well as an upper bound for the fractional chromatic number in cubic graphs with large girth.