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 
\[ P\{\text{the paceship gets into Solar system }\}\leq r/R.\]
You can find one proof [here][1]

   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 Matingale/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".
 
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  [1]: http://www.matapp.unimib.it/~fcaraven/did1011/mi/enterprise.pdf