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,
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&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 $s_i \gg r_i$.

  [1]: https://i.sstatic.net/RvcHl.jpg