TITLE: Set theoretic forcing, large cardinals and probabilistic methods
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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.