[jason1990]

I think I see a connection. Let Y be your answer (0.146) and let M be my answer (0.427). Then M = (1 - Y)/2.

I claim to have calculated M = the fraction of all the integers that the frog will eventually HIT. I see I misread the original problem on the IBM website. So my answer should be 1 - M = 0.573 = the fraction of all the integers that the frog will eventually MISS.

You have calculated Y = the probability, starting at 0, that the frog misses state k, for k near +infinity. Hence, Y is the fraction of states above his starting point that he will miss. However, since he will reach a bounded minimum, the fraction of states below his starting point that he will miss is 1. So the fraction of all the integers that he will miss is 0.5 + Y/2 = 1 - M.