The reward is 50Bitcoins?!
Thats 500-700 * 50? 25,000 - 35,000?!
How long must a miner mine before breaking the code?
Heh, unless you have a pretty great GPU/dedicated hashing chip setup you'll likely be paying a good deal in electricity costs to keep it running.
From memory there are a limited number of 'blocks' that are created on the network each day, these are the collective virtual receipts of transactions across the network during a certain period. In order to be valid they each require users who have opted in (miners) on the network to validate the blocks by solving a hashing puzzle.
The likelihood of a regular computer on the network successfully guessing the hash puzzle is less than once a year (likely even less now). What 'miners' are doing is essentially buying more and faster guesses with their mining rigs. So although theoretically a slow computer has a slim chance it would likely be guessed faster by others first.
I should add that I've read of pooled groups of miners collaborating, that way they share in the profits of the entire group. Haven't looked much into that side of it, but I'm guessing there would be minimum power requirements.