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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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Brownian scaling follows from the fact that the square of a simple random walk minus the number of steps is a martingale. – Steve Huntsman Jan 6 '12 at 16:31
community wiki? – Kevin O'Bryant Jan 6 '12 at 19:28
Martingales, Brownian motion, diffusions and percolation are some of the major workhorses in contemporary probability theory. This question is essentially, "Interesting applications of probability theory?" which is absolutely too general for the site. I recommend you reformulate this into a much more precise question. What specific applications do you have in mind? – Tom LaGatta Jan 6 '12 at 22:30
@Tom LaGatta "...Any question of interest to a wide class of mathematicians (such as this one) has a right to be posted here. – Tom LaGatta Jan 15 2010 at 22:11" :):) – Alexander Chervov Jan 7 '12 at 17:28
The thing that makes the question puzzling is that while there are fairly natural connections between martingales, Brownian motion, and diffusions, percolation is really apparently unrelated to the others (which is not to say that, for example, one cannot find martingale techniques used in percolation theory...) – mfolz Jan 8 '12 at 8:32

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!

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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:

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.

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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).

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