This makes mining simply a lottery. Every next attempt has the same chance of winning as the previous and the only thing you can do to increase your odds is increase the number of attempts you can make per second (= the hashrate).
No. Other than optimizing the speed at which the miner can makes its attempts, there is no way to prime or optimize the process.
I've written a fairly elaborate, but not too technical summary of the how and why of the mining process on my blog. If you're interested, you can check it out. The relevant posts are:
http://bitcoin.rannasha.net/?p=8 (On why we need mining and the blockchain)
http://bitcoin.rannasha.net/?p=23 (On what hashing actually is and how the mining process works)
http://bitcoin.rannasha.net/?p=29 (On pool-mining)
Too tired/lazy right now so I bookmarked for later.