We initiate the study of Local Computation Algorithms on average case inputs. In the Local Computation Algorithm (LCA) model, we are given probe access to a huge graph, and asked to answer membership queries about some combinatorial structure on the graph, answering each query with sublinear work. For instance, an LCA for the $k$-spanner problem gives access to a sparse subgraph $H\subseteq G$ that preserves distances up to a factor of $k$. We build simple LCAs for this problem assuming the input graph is drawn from the well-studied Erdos-Reyni and Preferential Attachment graph models. In both cases, our spanners achieve size and stretch tradeoffs that are impossible to achieve for general graphs, while having dramatically lower query complexity than worst-case LCAs. Our second result investigates the intersection of LCAs with Local Access Generators (LAGs). Local Access Generators provide efficient query access to a random object, for instance an Erdos Reyni random graph. We explore the natural problem of generating a random graph together with a combinatorial structure on it. We show that this combination can be easier to solve than focusing on each problem by itself, by building a fast, simple algorithm that provides access to an Erdos Reyni random graph together with a maximal independent set.