terminology and conceptual question about progress of PoW what is wrong about thinking 


  prbobability of attacker increases with time t that goes into PoW since they have more time to secretly mine their alternate blockchain ?


This refer to double spending. Can one think in these terms or is progress of finding a solution to pow to related to it. 

Please enlighten.
Time only helps the attacker if they have more hashpower than all other miners together. That scenario is called a majority-attack and discussed on numerous questions on this site. Such an attacker can essentially rewrite arbitrary lengths of history.

However, if the attacker only has minority hashrate, time works against them. The more time passes, the more likely is it that other miners advance the blockchain further than the attacker in the same time. Once the attacker falls behind, nobody will consider their chaintip when it is broadcast and the attack is failed.