The Probability to Hit Every Bin with a Linear Number of Balls

Assume that $2n$ balls are thrown independently and uniformly at random into $n$ bins. We consider the unlikely event $E$ that every bin receives at least one ball, showing that $\mathrm{Pr}[E]=\mathrm{\Theta }(b^n)$ where $b\approx 0.836$. Note that, due to correlations, $b$ is not simply the probability that any single bin receives at least one ball. More generally, we consider the event that throwing $\alpha n$ balls into $n$ bins results in at least $d$ balls in each bin.