[ilya_m]

We are talking about different kinds of "probabilistic" proofs. Igor Pak's blog post and the root message in this thread refer to statements that are validated only probabilistically. The probabilistic method, by Erdos and others, is a proof as rigorous as any. "Probabilities" in the this method are just a convenient apparatus for counting objects.

(As an aside, there's important distinction with statements that are believed to be true based on empirical evidence, such as hardness of factoring or the Riemann conjecture. If the bunkbed conjecture was refuted based on Monte Carlo simulations, it'd be possible to reduce the level of uncertainty arbitrarily low by throwing more resources at running more simulations.)

[bubblyworld]

Right - I admit I stopped reading the blog post at the point they linked to the paper and started reading that instead (which contains a rigorous proof). Having read the rest of the blog now I can see where the OP is coming from.