This question is an extension of the already fairly well discussed problem Does the drunk man fall off the cliff?
A few people raised this question of why the probability should not be 1, i.e. why does the drunk man not always fall off given he has infinite steps to take.
As quoted by a user in his answer.
If allowed to randomly step indefinitely means he keeps stepping until he falls off the cliff.
Some arguments against this were:
- "There are many infinite sequences of steps which never cross the cliff."
- "So If you could somehow collect an infinitely large set of computers and run a simulation, I maintain that a nonzero fraction of them would run without halting forever!"
To this I would ask: what about the case when q < 1 / 2. In that case the agreed upon answer is 1. So aren't there still infinite sequences of steps which never cross the cliff?
Of course there are still infinitely many sequences never crossing the cliff, but they make up a set of measure zero in the set of all sequences.