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A Simple Look at Simple Random Walks
A drunken man always returns home, but a drunken bird will eventually be lost.
Let’s say you’re standing on a number line at 0. You have the choice to either go left (in the negative direction) or to go right (in the positive direction). You’re equally torn between the two options, so the probability that you go either direction is one-half. You flip a coin — heads you go right, tails you go left. It comes up heads, and you move one step to the right. Now, standing at 1, you suddenly get amnesia. All you see is the coin in your hand and the choice to either go left or to go right. You flip the coin, and it’s heads again (but of course, you don’t know that this has happened before, with the amnesia and all), so you decide to go right. Standing at 2, you get amnesia yet again, and you have to now decide whether to go left or right, without any of your previous knowledge. Repeating this process, we see an exhibition of a one-dimensional simple random walk.
A Simple Random Walk in One Dimension
If I were to observe you doing this, I’d have some questions (aside from the obvious — why):
