user18783: Right now the network difficulty for mining is 40640955016.576. And, according to blockexplorer.com, the current odds of finding a block for any given hash is 0.0000000000000000000057288784391318092718842084135055792. 
My question is, how is the "difficulty" and the odds of finding a block related? Obviously a difficulty of ~4 billion does not mean there's a 1 in 4 billion chance of finding a block (any modern ASIC miner could find a block in a heartbeat). So what does it mean exactly and how does difficulty and odds relate? And how can one calculate the odds based on the difficulty?

user120: The difficulty is calculated in multiples of difficulty 1, which is defined as a probability of a single hash to be valid of 1/(2^32), i.e., in expectation one PoW is found by iterating through all 2^32 possible nonce values.
So given a difficulty of 40640955016.576 we have a probability for a single hash to be a valid PoW of 1/(2^32*40640955016.576). This in turn results in the ~5.72e-21 given by blockexplorer.

18783: <p>Right now the network difficulty for mining is <code>40640955016.576</code>. And, according to blockexplorer.com, the current odds of finding a block for any given hash is <code>0.0000000000000000000057288784391318092718842084135055792</code>. </p>

<p>My question is, how is the "difficulty" and the odds of finding a block related? Obviously a difficulty of ~4 billion does not mean there's a 1 in 4 billion chance of finding a block (any modern ASIC miner could find a block in a heartbeat). So what does it mean exactly and how does difficulty and odds relate? And how can one calculate the odds based on the difficulty?</p>
 120: <p>The difficulty is calculated in multiples of difficulty 1, which is defined as a probability of a single hash to be valid of 1/(2^32), i.e., in expectation one PoW is found by iterating through all 2^32 possible nonce values.</p>

<p>So given a difficulty of 40640955016.576 we have a probability for a single hash to be a valid PoW of 1/(2^32*40640955016.576). This in turn results in the <a href="http://www.wolframalpha.com/input/?i=1%2F(2%5E32*40640955016.576)" rel="nofollow">~5.72e-21</a> given by blockexplorer.</p>
