Suppose we have a point in 2D space, and a circle with radius $R$ a distance $D$ from the point. If someone was to move out from the point at an arbitrary, uniformly distributed, angle in a straight line forever, what are the odds, as a function of $R$ and $D$, the path chosen would intersect with the circle? I got an equation through geometric construction, but I'm not sure it's accurate.
$P(R,D)={R\over{\pi D}}$
I created the image attached to illustrate my thoughts. The red circle represents the point, and the blue circle represents the circle.
Preferably, I'd like a solution that generalizes to 3D and potentially non-circular objects.
The rays that just contact either side of the small circle will be tangent to the small circle. The angle between these two rays will be $2*\sin^{-1}(\frac{R}{R+D})$, using Pythagorean Theorem. So, the probability is given by $P(R,D) = \frac{\sin^{-1}(\frac{R}{R+D})}{\pi}$.