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 [E] = Θ(bⁿ) where b ≈ 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 α n balls into n bins results in at least d balls in each bin.