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I had this thought in my head for a couple of months now and I really wanted to see what the answer to it is. So here it is:

A man is sitting in a white room, where all he has to do is cross over a purple line to travel back to Earth. However, he has infinite time to do so. Does this mean he has a 100% chance of crossing the line, because he has as much time as possible, meaning that soon enough sometime he will cross the line?

(Also not accounting for eating, bathroom, or sleeping)

Display Name
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  • It depends on what he does. If he always sits still he will never cross the line. – Rammus May 28 '20 at 08:05
  • But the time he has is infinite, at some point he has to walk past the line. – Display Name May 28 '20 at 08:08
  • What is the definition for crossing the line? Does it count that, after millions of years, all of the atoms in his long-dead body would have displaced, with Brownian motion, so that the atoms are on the other side of the line? – Matti P. May 28 '20 at 08:14
  • I should have said that he would last forever... but I would assume no he would not die nor his atoms would be displaced. – Display Name May 28 '20 at 08:39
  • @5Flux How can he walk past the line if he never moves? The answer depends on what dynamics we give to the man. As Jordan Mitchell Barrett says, if we assume the man walks randomly then we can compute the probability that he crosses the line.

    However there are still many scenarios wherein the man will never cross the line. Suppose the line is one meter to the right of the man and he walks $\frac{1}{2^n}$ meters to the right on day $n$ (starting counting from day $1$). Then the man will get arbitrarily close but will never cross the line.

     – Rammus May 28 '20 at 08:43

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This can be answered somewhat precisely using the theory of random walks. We could model the man's motion as a random walk in the 2D-lattice $\mathbb{Z}^2$, starting at the origin $(0,0)$. It turns out that in one or two dimensions, the probability of eventually passing through an arbitrary point $(a,b)$ tends to $1$ as time goes to infinity. Interestingly enough, this doesn't hold in three dimensions, where the probability of reaching arbitrary $(a,b,c)$ is only $\sim 34 \%$ on average, and this decreases the more dimensions you add.

So your hypothesis is correct - given enough time, the man would eventually cross the line. Of course, this assumes the man in question is constantly moving randomly, which is a questionable assumption in itself...

Jordan Mitchell Barrett
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