I wanted to bring up the gambler's fallacy several pages ago, but I didn't have an account and so had to wait a while.

It seems to me there is a fundamental misunderstanding of how this whole hashing thing works. It is based on the erroneous belief that during each block you are making "progress" towards a share; progress which is discarded if a new block shows up before you actually find a share. This would be as though you were trying to fill up buckets, but someone kept taking the buckets away and replacing them with empty ones before you could ever fill one entirely. This is wrong. There is no progress.

The reason it's called the gambler's fallacy is this: Imagine a man at a casino. He's been at the same slot machine for hours without getting back a dime. Due to the amount of time and money he's already put in, he's convinced it has to pay off sooner or later and that quitting beforehand would mean forsaking his progress. If for some reason the casino had a weird policy wherein every five minutes he had to move to a new machine, he might be upset by this and claim that every time he moves to a new machine he loses all his "progress" on the previous one. In reality, as long as the different machines are mechanically identical, there is physically nothing different from pulling the crank on one or the other. A slot machine has no memory and has no way of "knowing" that you've tried 500 times and deserve a break.

As for mining, say blocks come by every 10 seconds and you average one share every 12 seconds. Based on a misunderstanding of averages as being discrete times at which a share will be found, one might conclude that he would never get a share as his "progress" would be "reset" 2 seconds shy of finding a share each time. In reality, he has the exact same chance of finding a share every single second, it just averages out to once every twelve. However, it wouldn't be too uncommon to find one in only 8 seconds, or have to wait 16. In fact, while very unlikely, it would be possibly for a CPU miner to solo-mine Bitcoin and find a block within seconds, while a guy with a top of the line ASIC doesn't find anything in a month.

In closing,consider the following: If you throw a die once, what are the odds you roll a six? One in six, right? Yes. How about if you throw it twice? 2/6? Nope. "What!?" you might say "But two rolls should clearly give me twice the chances as one, you're obviously an idiot." Yet by that logic, six throws would give you 6/6, a 100% chance of success, and life is never that certain. The answer is that two rolls net you an 11/36 chance. Why? The answer is actually quite simple: there are a total of 36 equally probable combinations of two rolls, 11 of which include at least one six ([1,6] [2,6] [3,6] [4,6] [5,6] [6,6] [6,1] [6,2] [6,3] [6,4] and [6,5]) And six rolls? Surely in six rolls you should get a six once right? In fact you only have a 66.5% chance. Still more often than not, but not exactly something to bet your life on.