There was a recent thunder storm in my city and it caused me to start thinking about a certain problem (perhaps it's my own, perhaps others have thought about it or some variant previously, I do not know). I have not come up with a solution(indeed, I scarcely have a clue how to), but I am sufficiently interested in this question to ask the brilliant minds on this site.
One remark before asking: if the question is vague mathematically in any sense and can be made more precise without seriously changing the substance, please feel free. On the other hand, if it is trivial, and I have missed the triviality, perhaps make it more interesting. This is very "informal"-ish but I think the gist of it is clear.
Suppose a man (whom we can view as an idealized point particle) is running around in a rectangular soccer field (width $w$, length $l$) at a constant speed $s$ for some period of time $T$. Let's suppose he starts at the centre of the field. Moreover, he is running "erratically": any "paths" of length $sT$ starting from the center are equally likely.
There is a thunder storm—there is a possibility he may be hit by lightning. Every $t$ unit times ($t<T$, of course) a lightning bolt strikes some point randomly in the soccer field; it has a "kill radius" of length $r$. The first bolt strikes the instant the man starts running.
What is the likelihood, in terms of the variables given, that he survives during the period of this run?
If the man can get within $r$ of the boundary of the field, things will be complicated and you'll have to be more precise about what you mean by all "paths" of a certain length being equally likely. But if the man can't get within $r$ of the boundary of the field, or if you're willing to neglect the boundary effects, the answer is quite simple: It doesn't matter where the man is; each lightning bolt has a probability $\pi r^2/lw$ of killing him. There are $\lceil T/t\rceil$ lightning bolts, so the survival probability is
$$\left(1-\frac{\pi r^2}{lw}\right)^{\lceil T/t\rceil}\;.$$