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Comment by ars | Hacker News Reader

This not true:

"Because the outcome of each coin flip is independent of all the others, there is always an equal chance that it will land on heads or tails with each toss. That means that your future final position is independent of your original starting position"

Assuming a fair coin, you will most likely end up approximately at where you started, with a Gaussian distribution. And your final position is extremely correlated with where you started.

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But once you move to 3-dimensions, with a 3-sided coin that isn't true any more.

Yes it is true. You may be misremembering the fact that an infinite walk in 3D has less than 1 probability of returning to start. But returning to start is still the most likely outcome. How could anything else be more likely?

https://mathworld.wolfram.com/RandomWalk3-Dimensional.html

They definitely chose their words poorly in that sentence.

In theory, using spherical cows, sure. But in practice there are all sorts of things screwing up the "probability" when jumping. Slope of the ground, windspeed, ground cover, etc.

Then it's still dependent on some known inputs: final position depends on initial position, wind speed/direction, ground conditions and gradient, etc etc. There is randomness, but you could make a probability distribution.

Ok? But it won't just converge to 0,0 because everything isn't equal and opposite, there will be some sort of bias that will move the bean away from it's origin.

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