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: $<x^2> = A \times t \space \frac{cm^2}{s}$, where $A \in \mathbb R^+$ is some positive real valued constant (and "cm" stands for "centimeter" and "s" stands for "seconds", just to use standard units).

What is the probability that the particle moves some distance $\leq k$ cm from $p_0$?

Also, if 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 without the reflecting plane, 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?