A well known interesting fact is that if two integers are picked "at random" (in an appropriate asymptotic sense), the chances they generate the integers is $6/\pi^2$. So, the integers can be generated by two randomly selected elements with non-vanishing probability.
I am wondering if there exists a (finitely generated) group that is almost certainly not generated by any finite number of "randomly chosen" elements. That is, is there a group $G$ with generators $\{g_1, ... , g_n\}$ such that for every $k$, our if we pick $k$ elements of $G$ at random, there is a vanishingly small probability that the selected elements generate $G$.
To clarify, select elements randomly from balls of a given radius in the group, using the word metric with given generators, and see if the probability $k$ elements chosen from the ball generate $G$ tends to $0$ as the radius of the balls grows. This definition is compatible with the above result about the integers.
from Hot Weekly Questions - Mathematics Stack Exchange
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