I did a little bit more Mathhammer for the HP odds of turning to stone, and since my first attempt was pretty much some back-of-the-envelope noodling, here's an update with a real probability formula:

The general formula for determining the odds of independent events (i.e. dice rolls) on multiple trials can be found with the following formula:

[N!/(N-M)!*M!] x p^M x [(1-p)^(N-M)]

Where N is the number of attempts, M is the number of successes, and p is the probability of the independent event.

I've assumed 12 casting attempts, using a 6 sided die (duh)

The odds of zero 1's : 11.22%

Odds of exactly one 1 : 26.92%

Odds of exactly two 1's : 29.61%

Odds of exactly three 1's : 19.74%

Odds of four or more 1's: 12.52%

In other words, there is a 32.26% chance that a HP will roll at least three 1's if he casts twice in each magic phase of the game. If he takes the Arcane Apparatus, there is a 12.52% chance that he will roll at least four 1's.

Just wanted to get that out there since my first calcs were more imprecise.