Jorge has lost his car keys! He's not using a very efficient search; in fact, he's doing a random walk. He starts at 0, and moves 1 unit to the left or right, with equal probability. On the next step, he moves 2 units to the left or right, again with equal probability. For subsequent turns he follows the pattern 1, 2, 1, 2 etc.

His keys, in truth, were right under his nose at point 0. Assuming that he'll spot them the next time he sees them, what is the probability that Jorge will eventually return to 0?

Note: For purposes of the problem, assume that Jorge is not looking while he is merely passing through a point. For example, he will not look for his keys while he passes through 0.

Charlie:

I agree that the probability that Jorge will eventually land on 0 is 100%.   While I have trouble following your explanation, I think it is wrong, and that you are overestimating the length of time it will take him to land on zero.  Specifically, by combining his odd and even step together into a single step of size 1 or 3, I think you saying that he will only land on zero on an even step.

Does your estimate consider that he might return in just 3 steps:  For instance: Left 1, right 2, left 1?  Why is the smallest number in your  simulation a 4?

Or maybe you have it right, and I have no idea what you are saying?  If so, could you explain differently?  Thanks,

Your respectful fan,

Steve

Posted by Steve Herman on 2009-01-18 00:22:45