Problem 46 Solution

Think of this problem instead as a random walk along a number line starting at the point 0, moving one unit to the right or left each turn. What if the there were barriers at points x+1 and -x-1, such that x and -x were as far as you could go, in which case your next turn would be one step closer to the origin?

Next determine the expected number of turns given such barriers using the same method of interlocking expected values as in the ant and spider problem. The answer turns out to be 2x.

Thus, when x is infinity the expected number of turns must be 2*infinity=infinity.

Related question: What is the expected number of turns it would take a random walker on a number line to revist any point? The answer is here.

Michael Shackleford, ASA