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\end{lstlisting}  The equation defines at which position the path will lie after $n$ number of steps.  \hspace{2000 mm} To model the distribution of $S_{n}$ we need to set it to $x$ and set the number of steps to the right and left by $r$ and $l$, respectively. Given this, $x = r - l$ and $n$, the total number of steps actually executed, $= r + l$. By simply solving for $r$ and $l$ we deduce that $r = (1/2)(x+n)$ and $l = (1/2)(n-x)$.  Given this, there are now $${n ${n  \choose l}$$ l}$  ways that l given steps can occur in a total of n steps. This also denotes the number of ways of arriving at point x (the end point), with each way having a probabilty of p^{r}q^{l}. One thing that must be noted is that x and n must be either odd or even. This is the general description of random walk itself. But the calculations to determine effectiveness of a shot falls under a statical analysis of binary classification called f1 score. It utilizes two terms - precision and recall. In terms of our mathematical model, precision is the number of tumor cells covered in the shot divided by the total number of cells covered by the shot. Recall, in context, is the number of cancer cells covered in a shot divided by the total number of cancer cells. The F_{1} score in our problem was calculated with this equation:  \begin{equation}   F_{1}=2\frac{(Precision)(Recall)}{(Precision + Recall)}