The probability that $r$ elements of a rank $n$ free group generate a rank $r$ subgroup

Granted the three integers $n\ge2$, $r$, and $R$, consider all ordered tuples of $r$ elements of length at most $R$ in the free group $F_n$. Calculate the number of those tuples that generate in $F_n$ a rank $r$ subgroup and divide it by the number of all tuples under study. As $R\to\infty$, the limit of the ratio is known to exist and equal 1 (see [1]). We give a simple proof of this result.