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?