Tigger: As you might know, there are some sites that allow you to review dice rolls after a game or series of games, showing what the actual distributions were--compared to the theoretical average values.  Neat stuff!  Sometimes I'll see that an opponent got more nice rolls than expected, but not so many that would clearly demonstrate some kind of cheating.  But those rolls and situations are painful...and they stand out in memory!

I think that in games involving dice (or in other games of imperfect information), very good players will combine a solid grasp of probabilities with an understanding of how those probabilities might unfold in play.  They'll do a better job of anticipating possibilities, and will tend to distribute their pieces in play so as to maximize positive outcomes and minimize bad outcomes.  I think that at times it can present the illusion that they're influencing rolls or getting access to hidden knowledge...