Imagine I have a point source $p_0 = (x_0,y_0,z_0)$ that releases a point-like Brownian particle with a lifetime given by an exponentially distributed rate parameter $\lambda$. When the particle's lifetime is over, it vanishes. We can also write an expression for the mean square displacement of the particle as a function of time as: $<x^2> = A \times t$, where $A \in \mathbb R^+$ is some positive real valued constant.

I place a ball of radius $r$ a distance $d$ away from $p_0$. What is the probability that (one instance of) the point-like Brownian particles released at $p_0$ hits and absorbs at the sphere before its lifetime is over and it vanishes? What if we release the particle on the surface of a reflecting plane (that prevents translation of the particle to coordinates where $z \leq 0$)? In the former case, is it possible to write down an exact analytic expression for the absorption probability, or is this generally too difficult to do for these sort of questions?