  It was recently shown in [4] that, for $L_2$-approximation of functions from
a Hilbert space, function values are almost as powerful as arbitrary linear
information, if the approximation numbers are square-summable. That is, we
showed that \[
  e_n \,\lesssim\, \sqrt{\frac{1}{k_n} \sum_{j\geq k_n} a_j^2} \qquad \text{
with }\quad k_n \asymp \frac{n}{\ln(n)}, \] where $e_n$ are the sampling
numbers and $a_k$ are the approximation numbers. In particular, if
$(a_k)\in\ell_2$, then $e_n$ and $a_n$ are of the same polynomial order. For
this, we presented an explicit (weighted least squares) algorithm based on
i.i.d. random points and proved that this works with positive probability. This
implies the existence of a good deterministic sampling algorithm.
  Here, we present a modification of the proof in [4] that shows that the same
algorithm works with probability at least $1-{n^{-c}}$ for all $c>0$.
