  We consider critical site percolation on the triangular lattice in the upper
half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in the
interior of the half-plane. It was predicted by Simmons, Kleban and Ziff in a
paper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2
\leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow
nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2
\leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where
$x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same open
cluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov proved
in a paper in 2012 an analog of this factorization in the scaling limit. We
prove, using their result and a generalized coupling argument, the earlier
mentioned prediction. Furthermore we prove a factorization formula for the
probability $\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw
\leftrightarrow [nu_1,nu_1+s])$, where $s>0$.
