Luck is a measurement of historical random results compared to their expected values. In Bitcoin there is no way to be 'lucky' in the future, no mining process will expect to give you better random results than the default expected results. The term 'luck' is generally misunderstood by most people, and anyone who thinks they 'will' have good or bad luck Bitcoin mining in the future, does not understand random probability. I use the term 'luck' here under duress due to the amount of misunderstanding of it's meaning.
Why is luck a major part of Bitcoin mining?
As mentioned on the Help->Mining page, Bitcoin mining is like rolling a dice. Finding a block wont happen just after every 100% Difficulty of hashes everyone on the pool does, it will happen randomly, but on average, after thousands of blocks, close to once every 100% Difficulty of hashes. So this means that sometimes it will not take very long to find a block, but sometimes it will take a long time to find a block. It's random.
Do we know what to 'expect'?
Yes indeed, we do know what to 'expect' when mining for Bitcoin blocks! The random properties of Bitcoin mining follow what is called a Poisson Distribution (link: here) After many results, it will look like a bell shaped curve, where in the middle it will be highest around the 100% Difficulty mark, and on the edges it will be lowest approaching the 0% Difficulty and 1400% and beyond Difficulty points on each side.
So what does this mean?
It means that we can say how often to 'expect' to find different Difficulty blocks as long as we have found a large number of blocks. So here's a table of Bitcoin block finding probabilites:
Difficulty
CDF
Probability Above
50%
0.3934693403
1 in 1.6
100%
0.6321205588
1 in 2.7
200%
0.8646647168
1 in 7.4
300%
0.9502129316
1 in 20.1
400%
0.9816843611
1 in 54.6
500%
0.9932620530
1 in 148.4
600%
0.9975212478
1 in 403.4
666%
0.9987188536
1 in 780.6
700%
0.9990881180
1 in 1096.6
800%
0.9996645374
1 in 2981.0
900%
0.9998765902
1 in 8103.1
1000%
0.9999546001
1 in 22026.5
1400%
0.9999991685
1 in 1202604.3
What this table means, using the 200% example, is that on average, over a large number of blocks, 1 in 7.4 blocks will be over 200% Yes that means that blocks over 200% Difficulty are actually expected to be quite common on every pool!
In the long run it all balances out since you also expected to get enough short blocks to make up for the less regular long blocks. However, since the process is pseudo-random, you cannot expect luck to change each time we get a long, or short block. Every block found is effectively independent of every other block found, so it will of course be random when luck will change from good to bad or bad to good. We may get a number of high Difficulty blocks in a row, or may get a number of low Difficulty blocks in a row, but we do not ever know in advance what the next block will be, and it doesn't matter what Difficulty the previous found block was.
Shares
Shares also follow exactly the same distribution, however there's way more of them, the average expected time per share is of course much shorter.
So to give an example with shares: A standard miner should mine, on average, 18 Shares per Minute, which averages to 25920 Shares per Day. Thus using the above table, a miner should find, on average, (25920 / 22026.5) Shares per Day over 1000% i.e. average 1.18 Shares per Day over 1000%. Now since a 1000% share would be 33.3 seconds, then on average 1.18 times a day, a standard miner mining at 18 Shares per Minute should take more than 33.3 seconds between finding shares.
But in all CDF probabilities, it's the size of the sample that matters, i.e. the number of shares. So we can extend the above table and the number of shares to give two other interesting numbers:
A 1 minute share is a 1800% share, with a CDF 'Probability above' of 1 in 65659969.2 So someone with 2533 miners would find, on average, that many shares per day. Thus they would, on average, have one miner of their 2533 miners, take 1 minute between shares, per day.
However, if we take it one step higher, to 2 minutes, i.e. a 3600% share has a CDF 'Probability above' of 1 in 4503599627370496 So even with a 1000000 miners, you would only expected to average 1 share over 3600% every 475.70 years. (4503599627370496 / 1000000 / 25920 / 365.25)
This explains why the Workers page highlights miners that have a time since last share of 1 minute as a warning and 2 minutes as an error. No one mining at 18 Shares per Minute should take 2 minutes to find a share, but with a lot of miners, may take 1 minute. It also shows that you really only need to check each miner every few minutes at the most since checking every minute will, on occasion, be a false positive if you have a few miners, but be a regular false positive if you have a lot of miners. However, after 2 minutes it is ever so unlikely to be a false positive even if you had 1000000 miners.
