  We derive an exact, simple relation between the average number of clusters
and the wrapping probabilities for two-dimensional percolation. The relation
holds for periodic lattices of any size. It generalizes a classical result of
Sykes and Essam and it can be used to find exact or very accurate
approximations of the critical density. The criterion that follows is related
to the criterion Scullard and Jacobsen use to find precise approximate
thresholds, and our work provides a new perspective on their approach.
