We will say gun 1 is located at a distance Xj from the collision origin and, when fired, will discharge a long bullet at same velocity Vp A similar arrangement exists for gun 2 using variable x2 , v2, For the sake of convenience we will state that when both guns are fired, then if Xj = x2 and vj = v2 the bullets identically collide. We would like to ask the question, if Xp x2, vj and v2 are some random variables (and the collision requirement geometry is known and kept simple) how do we calculate the probability the two bullets will collide. A particularly efficacious approach requires us to assign two new random variable q and p as follows: The distribution associated with p and q are still a type of convolution of the original variables (i.e., let 01 be the density function of xp 02 similarly related to x2; construct an x'2 such that x'2 = -x2 with an associated 0'2 , q - X] + x'2 and the density function of q is the convolution of 01 and 0'2 ). We construct a map of p and q; and outline the area where a collision may occur (refer to Figure 9):
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