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@HJRW: Thanks, that keyword gave me exactly what I want! The result I was looking for is contained in Woess, Wolfgang, Cogrowth of groups and simple random walks. Arch. Math. (Basel) 41 (1983), no. 4, 363–370.
@BenjaminSteinberg: I'm not sure I follow. Those seem to be about random walks on the Cayley graph of a free group, and the relevant probabilities seem to decay to $0$. Do you mean that we should write $G = F_n/K$ and think about whether a random walk in $F_n$ will land in $K$? I will have to think more about what exactly the answers are saying, but if they do answer my question it seems that they will give the answer "no" rather than "yes", right?
