I wonder if the probability is dependent only on $r_i$, or also dependent on the placement of $R$ within $S$? In these two examples, <hr /> <img src="https://i.sstatic.net/RvcHl.jpg" /> <hr /> it takes an average of $2.70$ steps to reach slicing $R$ on the left, but $3.16$ steps on the right. <br /><br /> I realize I'm ignoring your condition that $s_i \gg r_i$. <hr /> <b>Added 4Aug2020</b>. I include below some simulation data that might help a theoretical investigation. Here are two examples where $R = 0.2 \times 0.1$ in a unit square $S$. <hr /> <img src="https://i.sstatic.net/EOMW2.jpg" /> <hr /> On the left, after one million trials, the probability that the long side of $R$ is sliced was $0.591$. On the right, the probability was $0.622$. [1]: https://i.sstatic.net/EOMW2.jpg