This question is motivated by an exercise called "The Star-ship Enterprise's Problem" in Williams's book "probability with martingales", it can be stated as follows:

Suppose the control system on the spaceship has gone wonky. All that one can do is to set a distance to be travelled. The spaceship will then move that distance in a randomly chosen direction, then stop. The object is to get into the Solar system, a ball of radius $r$. Initially, the spaceship is at a distance $R(>r)$ from the sun. It can be proven with the help of martingale theory that the probability

$$P(\text{the spaceship gets into Solar system })\leq r/R$$ You can find one proof here.

So I wonder if there are some other examples in probability theory, they are interesting enough(of course interesting is an subjective manner) , can be easily formulated and understood by ordinary people, and are also nice applicaitions of Martingale/Brown motion/diffusion/percolation theory?

Here I add another well-known examples: The Equidistribution Problem in number theory, it can be solved by ergodic theory. It has a nice formulation as the reflection of a billiard ball on the table, see Hardy's book "An introduction to number theory".