We are given a rectangle $R$ with sides lengths $r_1$ and $r_2$, contained in a square $S$, with sides lengths $s_1=s_2\ge r_1$ and $s_2=s_1\ge r_2$. $R$ and $S$ are axis-aligned in a cartesian plane $P$. With the following recursive random process, we select straight lines orthogonal to the sides of $R$ (and $S$), until $R$ is cut.

At each time step, we select one of the two axes of $P$ with probability $\tfrac12$. Let $a$ the axis selected. Thereafter a straight line $L$ is selected uniformly at random from the ones cutting $S$ and orthogonal to $a$. Let $S'$ and $S''$ be the two parts of $S$ generated by the cut of $L$. These two random steps are repeated until $R$ is cut by $L$, and each time $R$ is not cut, $S$ is transformed by removing its part (either $S'$ or $S''$) that does *not* contain $R$.

**Question:** Given the coordinates of the vertices of $R$ providing its position within $S$, what is the probability $p_i$ that it is eventually cut (at the end of the random process) by a line orthogonal to its sides with length $r_i$ for $i\in\{1,2\}$?

(For the sake of clarity, we obviously have $p_1=1-p_2$.).

notpositioned in a corner of the square? $\endgroup$ – Penelope Benenati Aug 3 '20 at 17:31