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Joseph O'Rourke
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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,


      
it takes an average of $2.70$ steps to reach slicing $R$ on the left, but $3.16$ steps on the right.

I realize I'm ignoring your condition that $s_i \gg r_i$.
Joseph O'Rourke
  • 150.8k
  • 36
  • 358
  • 958