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 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 $\sum c_n^2<\infty$.

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 $\sum c_n^2<\infty$.