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 />
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/RvcHl.jpg" />
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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 $s_i \gg r_i$.
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<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 $S$.
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<img src="https://i.sstatic.net/EOMW2.jpg" />
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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