I'll play your game and assume for the sake of argument that the average occurs at set intervals - it doesn't at all, just because the guy gets an average successful guess of once per minute it in no way means that he was "going" to get it in ten seconds just because 50 had already elapsed - but even if it did, if it's just a guess anyways why would it be any less valid for the next card? He wasn't homing in on the correct number, his guesses are random. It is just as likely he will get a guess on the next card in those next ten second if we assume average=interval (which it does not).
Its an AVERAGE. Every time he he gets it quicker he necessarily gets it just as later another time. Otherwise it wouldn't be the average anymore.
Edit:
To your other point, about it not cointing towards not next "game" (block)
When a block changes, the whole thing resets. Everyone starts from scratch. He is once again up against the same probabilities, the same as everyone else. His guess will be useful for the next game, but
so will everyone elses. Just as his very next guess could be a winner, anyone elses could too... no advantage over anyone else.