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Probability Level 3

Cosmin is located at the origin of an x y xy -plane. Every minute, he will move up, down, left, or right by one unit. What is the probability that he will ever return to the origin?


The answer is 1.

Alan Yan by Alan Yan , 20, USA

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1 solution

Eli Ross Staff
Feb 12, 2016

He will, almost surely, return to the origin. (I encourage someone to post a rigorous proof of this!)

What's extremely interesting is that for 3 or more dimensions, the probability is < 1. <1. For example, in 3 dimensions, the probability is ≈ 34 % . \approx 34\%. See here for more details.

Can anyone make an intuitive argument for why the probability would be 1 in one or two dimensions, but then become < 1 <1 in higher dimensions?

4 Helpful 0 Interesting 0 Brilliant 0 Confused

The problem would be much more interesting if answers with decimal parts were allowed. This way it's just a choice between 0 and 1, and that's too obvious.

Marta Reece - 3 years, 2 months ago

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That's how I got the answer. Heehee

Mohammad Farhat - 2 years, 7 months ago

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