Double hashing - probe count probabilities
As Wandering Logic noted, there are approximate expressions for probabilities of probe count in unsuccessful search  case in TAoCP, a few pages later expressions for expected probe count:

$1-\alpha$, $\alpha(1-\alpha)$, $\alpha^2(1-\alpha)$, ... - probabilities of making 1, 2, 3, ... probes respectively.

In successful search case I derived approx. expressions myself, relying on explanation of expected probe count formula in the book:

$1 - {\alpha\over2}$, ${\alpha\over2} - {\alpha^2\over3}$, ${\alpha^2\over3} - {\alpha^3\over4}$, ...