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To begin, for the creation of a tumor, a mathematical phenomenon called random walk was utilized to formulate a tumor. Random walk is the mathematical formalization of a path that consists of a succession of random steps. Random walk sets, or takes independent random variables, denoted in our case by Z_{1}) where each independent variable has a value of 1 or -1 and an equal probability of .5. Additionally, it sets the sum of 0 steps (S sub 0) to 0 and defines (S sub n = blah blah) the sum of n steps as the sum of each random variable up to n. This essentially defines what the value will be after n number of steps.  To model the distribution of S sub n we need to set it to x and set the number of steps to the right and left by r and l, respectively. Due to the fact that most random walk distributions favor the right side marginally more, 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 choose 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. In the end, the probability distribution is described as:   BBLAH BLAH 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. This is denoted by 2 times (precision times recall)/(precision + 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 the t was calculated by taking the product of