## Saturday, August 14, 2010

### Random Walks

I found the discussion on random walks quite interesting. It is such a simple concept; I can’t believe it took Einstein to realise it is a satisfactory explanation for Brownian motion. I think this demonstrates the lack of statistical intuition that most people, even scientists, suffer from.

What I think people fail to appreciate is that in a process that is that consists of steps that are independent, the steps are genuinely independent of each other. Each step has absolutely no idea what the outcomes of the events preceding it were.
Say a person flips a fair coin five times, and gets tails every time. If we were asked to predict what the result of the next flip is, at least part of us would be tempted to say a heads is more likely. An average person might justify this thought as the coin “being due for a heads”. However, we know that the result of the next flip is completely independent of the previous flips. The coin does not know that it’s “due for a heads”. It has the same chance as it always did of resulting in a heads when flipped.
Assuming that the next trial in a series of independent events will be more likely to result in a value that brings the average of the results towards the mean expected outcome of all the events previous, rather than the known likelihood of the results of the event, is called the Monte Carlo Fallacy. When this fallacy is avoided, it is much easier to accept that random motions can result in a visible net displacement, if the observer waits long enough.

What I found even more interesting is that the random walk has structure on all length scales. The idea of an object with structure on all length scales is impressive. It is like a fractal. Like a fractal, whose total perimeter can be calculated, the length of a random walk can also be calculated. It seems necessary that the path of a molecule should have structure on all length scales though. Considering a single molecule in a very large sample of water, like a lake, or an ocean, or the air molecules in an atmosphere, then if enough time passed, the particle should have the possibility of being anywhere in the mixture. The position of the particle should not be restricted to a subsection of the space, because there is nothing to enforce that restriction. So the path traced by the particle has structure even on the scale of the size of the medium it is in. There should be no restriction on the length scale of the path of a particle, other than the size of the medium that the particle is in.

I suspect that the concept of the random walk has not been fully explored, and I think that this idea will be used again in future chapters.