I was reading "The Math Book" by Clifford Pickover yesterday and I came across an entry that i just haven't been able to make sense out of. I've been thinking about this since yesterday without it making sense and I figure maybe someone here would be interested or able to shed some insight.
"Imagine a robotic beetle placed in a twisting tube. The creature executes an infinite random walk by walking forever as it moves randomly one step forward or one step back in the tube. Assume that the tube is infinitely long. What is the probability that the random walk will eventually take the beetle back to its starting point?
In 1921, Hungarian mathematician George Polya proved that the answer is one - infinite likelihood of return for a one-dimensional random walk. if the beetle were placed at the origin of a two-space universe, and then the beetle executed an infinite random walk by taking a random step north, south, east, or west, the probability that the random walk would eventually take the beetle back to the origin is also one.
Polya also showed that our three-dimensional world is special: Three-dimensional space is the first Euclidean space in which it is possible for the beetle to get hopelessly lost. The beetle, executing an infinite random walk in a three-space universe, will eventually come back to the origin with a 0.34 or 34 percent probability. In higher dimensions, the chances of returning are even slimmer, about 1 / (2n) for large dimensions n. This 1/(2n) probability is the same as the probability that the beetle would return to its starting point on its second step. If the beetle does not make it home in early attempts, it is probably lost forever."
Basically I'm not understanding how the beetle could not find its way back given an infinite amount of steps.
Possibly this has something to do with orders of infinity? I'm only a little bit familiar with these, the first having the same cardinality as natural numbers. The only examples i can think of for second order infinity are sets of irrational numbers and the power set of any set of first order infinity. Perhaps the number of choices the bug has is less than the number of options available in 3+ dimensions? Even if this were so, it still seems like the bug should return to its original spot. Since there are infinite steps, i would think all points in the infinite space would be covered.
Speaking of infinity. If you were to flip a coin infinitely many times, not only would you get heads two-hundred million times in a row, but you would do this infinitely many times, correct?