  Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean,
unit variance and a bounded subgaussian moment. We show that the condition
number $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate
$${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c
t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment.
Although the estimate can be obtained as a combination of known results and
techniques, it was not noticed in the literature before. As a key step of the
proof, we apply estimates for the singular values of $A$, ${\mathbb
P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq
k\leq n,$ obtained (under some additional assumptions) by Nguyen.
