Is mining cumulative? Meaning if I wait those 500 days I should see a payout of the 25BTC or does everything start over once a block is solved? Excuse me if this is stated in Satoshi's original paper, I'm still working through it.
Mining is like rolling dice. (Ok, a die with 1.16 x 10
77 sides, but still, the analogy holds up pretty well).
Lets say you roll a normal 6 sided die. I set a difficulty saying that you "win" anytime the die comes up with a number less than 6. Most rolls you are going to "win". Then I decide that the difficulty needs to be increased. So, now I set a difficulty saying that you "win" anytime the die comes up with a number less than 4. You will "win" about half the time you roll. I can increase the difficulty further and state that you "win" anytime the die comes up with a number less than 2. Now on average you'll have to roll the die 6 times before you'll "win" once. It is entirely possible that you could roll a 1 a few times in a row. It is also entirely possible that you could roll 18 times without ever rolling a 1.
If someone else is rolling their own die, it has no effect on how long it will take you to roll a 1. They may roll a 1 sooner than you, they may roll a 1 later than you.
Now imagine that same die with a LOT more sides. Hard to picture a die like that in your mind, but essentially that's what is happening when you are solo mining. Your computer very quickly (millions of times per second) computes a hash that results in what is effectively a random number between 0 and 1.16 x 10
77. If the number is less than the target difficulty, you "win" and get the 25 BTC. If the number is not less than the target difficulty, then you "roll again". You just keep going until you are lucky enough to end up with a low enough hash.
Meanwhile, every other miner and pool is doing the exact same thing. Whether they happen to find a random hash that is less than the target (and how many times per second they can compute a hash) doesn't
directly effect your ability to do the same. If they happen to find one, everyone just creates a new block and continues generating their hashes.
Now, here's the catch. That "target difficulty" is regularly adjusted, and it is adjusted based on how fast hashes less than the current target difficulty are showing up. So as more and more people generate more and more hashes simultaneously, the odds of someone finding one increase (imagine 2 people each rolling 1 die, vs 10 people each rolling 10 dice. The number 1 will show up a lot more often in the second example). As the average time between hashes decreases, if it is less than 10 minutes, the target is lowered making everyone less likely to generate a hash value that is less than the target. So while their rolls do not change your odds of solving at the
current difficulty, they can influence future difficulties (it adjusts approximately every 2 weeks) making you less likely to "solve" a block in the future.
When someone says "it takes 2 years for you to find a block", that's a lot like saying "it takes 6 rolls to roll a 1". What they are really saying is that
on average if the difficulty stays exactly what it is today, you would be able to find 1 low enough hash for every 2 years of continuous hashing. Now, it is possible for you to find 2 or 3 hashes low enough, but it is also possible that you could go 6 years (or more) without ever finding a low enough hash.
Much like with dice, as you "roll more dice at once" your odds of hitting a target on each attempt increase. So, pools issue work to all their miners and get the benefit of all that hashing power increasing their odds of solving a block. When they do, they generally split it up proportional to the amount of hashing power each individual contributed to the effort. (The guy who rolled 2 dice gets payed twice as much as the guy who rolled only 1 die, and half as much as the guy who rolled 4 dice regardless of which of the 7 dice hit the target).
