Set theoretic forcing, large cardinals and probabilistic methods This question is motivated by the work of Ajtai "The complexity of the pigeonhole principle" and  similar works. In this paper, the author proves that $PHP_n$, the pigeonhole principle for $n,$ does not have polynomial-size constant-depth Frege proofs. The method of proof is an arithmetical analogue of forcing (of a kind already used by Paris and Wilkie), plus a  probabilistic argument to handle the relevant combinatorics.
Now my questions are the following.

Question 1. Are there similar works, which connect set theoretic forcing with probabilistic arguments in an essential way?
Question 2. Are there  works, which connect large cardinals and probabilistic arguments?

 A: Random real forcing is naturally connected with probability and measure theory, in an essential way, since the generic real that is added by the forcing has all the Borel properties that hold with probability one for reals in the ground model. 
One recent example is our recent paper on the rearrangement number (with six authors, available soon; I'll update with a link), where we combine probabilistic arguments and forcing. For example, in order to show that the rearrangement number was at least as large as the covering number for measure, we had considered the randomly signed harmonic series as used Rademacher's theorem that a randomly signed series $\sum (-1)^{r(n)}c_n$ converges almost surely just in case the series is square-summable $\sum c_n^2<\infty$. 
A: Today I saw the following paper in which probabilistic arguments are used in a forcing argument:
Halfway New Cardinal Characteristics.
See the proof of 3.4. The paper is written by Jörg Brendle, Lorenz Halbeisen, Lukas Daniel Klausner, Marc Lischka and Saharon Shelah.
