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We consider critical site percolation on the triangular lattice in the upperhalf-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a site in theinterior of the half-plane. It was predicted by Simmons, Kleban and Ziff in apaper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2\leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrownu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2\leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where$x\leftrightarrow y$ denotes the event that $x$ and $y$ are in the same opencluster, and $K_F$ is an explicitly known constant. Beliaev and Izyurov provedin a paper in 2012 an analog of this factorization in the scaling limit. Weprove, using their result and a generalized coupling argument, the earliermentioned prediction. Furthermore we prove a factorization formula for theprobability $\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw\leftrightarrow [nu_1,nu_1+s])$, where $s>0$.

Factorization Formulas for $2D$ Critical Percolation, Revisited