Return to Answer

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 $s_i \gg r_i$.

---

**Added 4Aug2020**. 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 $S$.

---

      

---

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 $0.622$.

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 $s_i \gg r_i$.

---

Added 4Aug2020. 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 $S$.

---

      

---

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 $0.622$.

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 $s_i \gg r_i$.

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 $s_i \gg r_i$.

---

**Added 4Aug2020**. 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 $S$.

---

      

---

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 $0.622$.

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 slice $R$ on the left, but $3.16$ steps on the right.

I realize I'm ignoring $s_i \gg r_i$.

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 $s_i \gg r_i$.