Interesting applications of [Martingale/Brown motion/diffusion/percolation ] theory?  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".


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 A: There are some more examples in Williams' book; my favorite is the "abracadabra" problem, which I state like this. 
Pick a random number in $[0,1)$, and looking at its decimal expansion, the expected number of digits you need to examine before finding the first "12183" is strictly less than the expected number to find "12381". Most everyone finds this surprising!
A: In computer graphics to generate textures like mountains, clouds,
one can use things similar to trajectories of Brownian motion. 
To my taste these pictures are quite nice:
http://www.gameprogrammer.com/fractal.html
Or search on "RMD = random midpoint displacement" algorithm - plenty pages on web.
Going into more mathematical details: in one dimension RMD can generate
franctional Brownian bridges.
However 2-dimensional process generated by RMD is not 2-d fractional
brownian motion (since it is NOT rotation invariant) however it might not be important.
A: The Gambler's Ruin Problem is a nice motivator for martingale techniques (the wikipedia solution is really a martingale solution in disguise, but not totally rigorous -- it can be made so by using the Optional Stopping Theorem for martingales).
