Invasion percolation is a "self-organized critical" distribution on random subgraphs of Z^2, believed to exhibit much of the same behavior as critical percolation models. Self-organization means that this happens spontaneously without tuning some parameter to a critical value. In two dimensions, some aspects of the invasion graph are known to correspond to those in critical models, and some differences are known. We will discuss new results on the probabilities of various "arm events" -- events that connections from the origin to a large distance n are either present or "closed" in the invasion graph. We show that some of these events have probabilities obeying power laws with the same power as in the critical model, while all others differ from the critical model's by a power of n.