  We prove a formula, first obtained by Kleban, Simmons and Ziff using
conformal field theory methods, for the (renormalized) density of a critical
percolation cluster in the upper half-plane "anchored" to a point on the real
line. The proof is inspired by the method of images. We also show that more
general bulk-boundary connection probabilities have well-defined,
scale-covariant scaling limits, and prove a formula for the scaling limit of
the (renormalized) density of the critical percolation gasket in any domain
conformally equivalent to the unit disk.
