Probability of missing cells when walking on the polyline
The total length of the polyline is \begin{equation} L=\sum_{i=1}^n L_i \end{equation} where $n$ is the number of intersecting cells, $L_i\in\left[L_\text{min},\,L_\text{max}\right]$ is the intersecting length for cell $i$, $L_\text{min}=\min_{\forall i\in[1,n]}{L_i}$, and $L_\text{max}=\max_{\forall i\in[1,n]}{L_i}$. Let $S$ be the step distance for walking. Assume that we only know $L_i$ for the current cell $i$, $L_\text{min}$, and $L_\text{max}$. If we knew all $L_i$'s, we could simply determine whether or not we would miss cell $i$ by checking \begin{equation} \sum_{j=1}^{i-1}L_j\leq S\left(\left\lfloor\frac{\sum_{j=1}^{i-1}L_j}{S}\right\rfloor+1\right)\leq\sum_{j=1}^i L_j. \end{equation}
- Formulate the probability of missing cell $i$ that actually intersects the polyline.
- What is the probability of missing any cells on the polyline?