If I'm attempting to mutate one arbitrarily chosen binary string $s_a$, to another arbitrarily chosen binary string $s_b$, in the smallest number of steps (i.e. with the smallest number of mutations) via a procedure where I decide to either:

(1) Randomly and with uniform probability choose a "0" bit and flip it to a "1" bit.  E.g. if there are three "0" bits in our string $s_a$ at some decision / mutation step, and $10^2$ "1" bits, here a particular "0" bit would be selected for mutation to a "1" bit with probability $\frac{1}{3}$.

(2) Randomly and with uniform probability choose a "1" bit and flip it to a "0" bit.

Is there a good intuitive reason why I might not want to always choose (1) or (2) based on which option maximizes the probability that the Hamming distance to the target string will be smaller in the next step?

Let me write a specific statement about where my thinking is failing:  I thought it would be the case that any string within a fixed Hamming distance of the target string would have a (topologically) indistinguishable representation on the $n$-dimensional hypercube where vertices represent all possible states of a length $n$ binary string (i.e. a length $n$ Hamming code - http://en.wikipedia.org/wiki/Hamming_distance).  Given that the $n$-dimensional hypercube is a distance transitive graph, all vertices within a fixed distance $k$ of another vertex should be identical up to isomorphism of the $n$-cube (http://en.wikipedia.org/wiki/Distance-transitive_graph).  Now, since "passing through" each set of topologically indistinguishable vertices within a Hamming distance $(1,2,...)$ of the target string *must* be done to reach the target string, what advantage could we have in not maximizing our chance to move one step closer to the target string at each step of the aforementioned decision process?

The above is NOT intended to be anything other than an illustration of where my intuition is likely failing.

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Note: This question is related to http://mathoverflow.net/questions/155076/the-time-to-drift-a-binary-string-from-one-state-to-another-via-deterministic-se.  I also asked this question in a long-winded manner in a (now deleted) question that received no answers, but where Douglas Zare in the comments claimed that a "Hamming distance" strategy for the above decision process failed in simple counterexamples.  I haven't been able to see this, it's on me and I thank him for his time.  But I'd really like to understand better what's going on, so I'd like to ask the above with the "soft-question" tag.