Topic: Vladimir's essential self-defence guide for Bitcoin Miners (Read 13274 times)

rfcpool is most excellent for a 24/7 miner imo

PPLNS FTW!

I don't know and I have no idea how to find out, so take the following figures as untested approximations.

If you look at other proportional pools, the increase from auto-hopping is in the order of 150 - 300 Ghps (correct me if I'm wrong). We could expect a similar amount from deepbit, making the hashrate increase from hopping only ~ 6%. There is no way this will reduce payoffs for fulltimers on prop by 50%.

If there are some miners who only hop from deepbit PPS to deepbit prop and back very few would be using an auto-hopper to do it (since so many more options are available).

Finally if you look at a graph of deepbit's hashrate over time any significant increase in hashrate that coincided with a found block and the start of a new round would stick out like a sore thumb. You don't see anything like that on deepbit hashrate graphs.

From what I've seen irl and in sim, the loss of income due to hoppers increases when the hashrate of the hoppers increases compared to the hashrate of the pool, and is only significant when the hashrate is a large fraction of the overall pool hashrate. I could be wrong though - if you think I am please explain why.

The PPS fee on Deepbit is way too high, that would just be stupid. Hopping is done to maximize profits, and you don't do that by using a pool with a 10% fee. The pools are envious of Deepbit's market share and lots of people thinks it's too close to 50%, so they will say anything to get miners to move to other pools.

Losing 50% of his BTC on short rounds at deepbit?  Sounds like someone doesn't know how to do math....  There would have to be a ridiculous number of hoppers for that to happen.  They may hurt some on shorter rounds, but at a giant pool like deepbit it should barely be felt.

That guy probably just had some bad luck with those short rounds where "he lost 50% from hoppers being in the pool".

I'll give your thread a bump, and summarize my points:

- Bitcoin block finding is random - pools may have runs of bad or good luck, but past results can't predict how a pool will perform in the future,

- To say a pool is cheating users based on luck is speculative, because even with many rounds of data, the statistics can't say that the probability is outside of expected variance. If I flip "tails" 10 times in a row in my coin-flip game, how do I then defend against allegations of impropriety just because it is unlikely? If my pool has 10 unlucky rounds in a row, should users be entitled to accuse me of stealing their earnings?

- Cheating by pool operators is hard to detect. Many mechanisms could be employed, such as only withholding very lucky block finds, diverting some user's shares to a second 'dark pool', adding non-existent shares to the pool to get paid for work not done, etc. Improving mining clients can help - it is possible for a miner to know that they submitted a winning share of higher difficulty than the bitcoin target, and the submission of a winning share can be locally logged; when the pool op knows that users know when a block was found, cheating is harder.

- When calculating "luck", even for curiosity's sake, one must convert the shares per round to a percentile or sigma of expected length, using good math aware of the true Bitcoin probability underlying pool shares, otherwise you will have skewed findings.

- Losses from pool hoppers is not hard to detect. Is the pool's hashrate graph jumping up at the start of every new round? In a proportional pool, your round earnings should be inversely proportional to round length. When you find that you are not making as much per submitted share as you should in short rounds, and are also able to submit fewer shares than you should have been able to in short rounds, it's time to jump ship.

I think you should read my post again. I was showing deepceleron how it could be possible to look at variation in pool luck using shares per block.

I have never mentioned wanting to include 'irrelevant' variables.

I never mentioned that I was not happy with your efforts.

I also have never made any indication that number of blocks found in a time period is a good indicator of pool luck, when average shares per round seems a more easily calculated index. So, no, I won't do calculations based on blocks per unit time. I can calculate for you the number of difficulty 1 hashed expected to find a block, but I'm hoping I don't have to. 

Was this just to get your thread to the top of the forum? Shame on you!

Without having precise details and reports on failure dates, times, and durations for each pool its impossible to do anything other than lump them in.   As long as this is done for ALL pools, its fair, isnt it?  The point im trying to make is that removing REAL occurences that affect availability and earnings only because they cant be predicted or determined to be recurring isnt really the right way to determine where to mine (or not to mine) either.   Your method suggests to completely abandon meaningful data simply because it may be phenomena (at least if if understand you correctly).  It could also be incompetence, you know (which does tend to be recurring and probably even with a known distribution)

He only compared one pool to the perfect record. We only have his word that btcguild is significantly worse than the other pools in this comparison. He refuses to post the data. For all we know, btcguild could be middle of the road compared to other pools as is shown at http://www.l0ss.net/index30.php.


Software, hardware, and network failures do not follow the binomial distribution of solving bitcoin blocks. Lumping them in as if they were recurring phenomena subject to a known distribution is just pseudo-math. The point is that you don't care to understand why, and that is fine. Feel free to hash where ever you like, for whatever reasons you like.


Glad you guys managed to scare up some business.

You could also run a quick simulation (see https://bitcointalksearch.org/topic/m.486238) and do a visual comparison on real data, which yes, you'd need to renormalise for each difficulty period. The advantage of using shares per round as a measure of luck is that renormalising the data is quite simple (list of difficulties and change dates here)

Note that this wont prove anything, but if you can model your simulation data accurately enough you'll be able to measure how far the real world data varies from it.

There is a lot of stuff i don't understand in this thread, in Vlad's defence i might say that he kinda makes it a little bit easier for me to digest and this thread made me stop trusting blindly in pool ops.

what do you guys have to say about rfcpool, it's not in the list. thanks

Wasnt that the whole point of this exercise anyway (and the original post for that matter?) 

And if not, why not? 

What do I,  as a miner,  care about your hardware, software, or network failures?  I want a return on my electricity costs + margin and the rest is your problem and why i pay you a fee/donation.  I dont care about normalized statistical calculations about the probability of x and y happening if the end result is the pool  solving less blocks than expected or hoppers stealing a share of my earnings.

Judging from the growth we have seen at Mainframe,  im guessing we arent the only ones with this viewpoint.

I think we need to remember that whether or not Vladimir's math and methods are 100% agreeable to everyone,  the basic point of what he was trying to say is valid and sound, good advice for miners.  Beware pools which appear to under earn and pools which allow hopping (unless you plan to hop yourself).   Not because they are evil or cheating or anything else (while its possible they could be) but because it eats into your earnings.  This is just common sense and good advice and all the math in the world cant replace it.

There are ephemeral effects that can take place which can degrade a pools ability to win this race. They can be both external and out of the operators control (DDoS or a macro-network failure) or internal and under control (such as a subsystem failure or degradation). Modelling the probability of these events is difficult since they are usually large and have few recurrences. It is also not clear what the impact for any particular event was on any particular pool. Also, a software patch applied to some pools but not others could degrade the ultimate process of obtaining the 50BTC without anyone quantifying the extent of the degradation or the period of occurrence at all.

My point was that since block finds can collide, we cannot assume the events are independent. Nor can we assume that the variance of that dependence can be known at a given point of time with any certainty. Rather than supposing we understand how the distribution of rewards for found blocks may vary from one that is appropriate for Poisson, we can use a different method and be certain.

However, it may not be necessary to model that. Instead we can look at how close each pool comes to the expected block solves for a perfect pool with zero hardware, software, and network failures. Then we can have a degree of confidence about expected performance in practice. This would be a very annoying data collection however, given that not all pools provide the data. If only we knew someone who had already compiled 3 months worth of data from the other pools and could post that data on the internet. Sadly, no such figure exists. The closest I have been able to find is at http://www.l0ss.net/index30.php. Of course, this analysis would only be useful as hindsight and have little in the way of predictive powers.

One thing we can be absolutely certain of: no matter how many people explain to Vlad how his conclusions are flawed, he won't change his story in the slightest. The narrative is likely to increase the size of his pocketbook and thus have "zero variance". 

deepceleron already answered that. Including data from multiple difficulty changes complicates things.

I don't feel like busting out my stats, but just a visual examination shows difficulty 567269 to look really bad. I think an analysis of that period whould show a significant deviation from expected results. Something happened to BTCGuild between 6/6 and 6/15.

OK, got it. This ->


is a bit of a broad statement and confused me a bit.


Edit: This is totally off topic, and you all have my apologies, but why isn't


suitable for the same job? I thought that was what Vladimir was doing and I'm not sure I understand your reasoning behind not using it.

The binomial cumulative distribution function differs depending on the number of trials. This is the source of my "interest rate" analogy.

Bitcoin probability of a single hash solving a block is 1/(Difficulty*2^32). Pool statistics obfuscate the actual number of hashes by being a 1 in difficulty problem wrapped around a 1 in 2^32 problem.

The function (1-(1/Dif))^(total shares) does not have the same quantiles as (1-(1/(Dif*2^32)))^(total hashes).

It is possible to look at the histogram of number of shares in a round for a large number of rounds to get a visual idea of pool luck luck. Further, you can calculate expected round length by total shares in a round. From Meni Rosenfeld's excellent work in progress:


and the likelihood of a round being solved but a certain number of submitted shares is given by the CDF (using R)

for 100000 shares:
> pgeom(100000,1/1777774)
[1] 0.05469788

for 10000000 shares:
> pgeom(10000000,1/1777774)
[1] 0.9963935

So a round has a 5.5% likelihood of being solved by 100000 shares, and a 99.63% likelihood of being solved by 10000000 shares.

This is not a criticism of your long term approach, I haven't yet read through it all. But your opening paragraph does leave you open to misinterpretation.

Holy crap, don't quote my whole post again!

Invalid block solves or orphaned blocks, of which there was one in the period I showed above, is a completely different mechanism than the statistics of block finding. If your pool finds a block solution, but another pool finds another essentially simultaneously, the winner is the one who gets it published and distributed on the p2p network first and fastest, so that future block hashes are built on it. You don't get the 50BTC if your block is not included in the final blockchain. This generally means that the non-orphaned solution is the one found first, but it is also could be that the ultimate winner had higher connectivity and was able to get their solution to more p2p nodes faster. This is hard to statistically model, but we could at least say that if pool 1 had five orphaned block solves (which they can publish to prove that the invalid was actually found within seconds of the winner), and the block solve winner was always pool 2, we might suggest that pool 1 should optimize their connectedness or miner work pushes to get their block finds out sooner on the Bitcoin p2p network.

I could run a exact confidence level model on several months of block finding, but I'm not enough of a math genius that I can immediately see how to also model the changing probability (difficulty) for the N trials in three months in code. It would probably involve making a data frame of all the block solves and running the vectors of shares and round difficulties through the poisson test though. We can see from my second code snippet above, with 100,000 difficulty 1 block solves (many more than the 2275 rounds) at a probability 1/2^32 (much lower than 1/difficulty) we still have a +/- 1% confidence window. Besides, it is improbable, but not impossible, for a pool to never solve another block in your lifetime.

@Vladimir
Numerous people have, in multiple threads. You just fail to learn from your repeated schoolings.
More appropriate treatment of your dataset has shown repeatedly in multiple threads that BTCguild performance is well within the expected behavior.
Expanded datasets that include other pools have shown that you are completely off base. You deny datasets larger than your own as "insignificant" in attempts to warp the data to fit your hypothesis.

You keep pasting your tired old story, and it never stops being wrong. You keep making yourself look foolish. At this point, anyone who trusts you with their bitcoins is even more foolish.

The first problem with your statistics is you are just using shares in your calculations. This is not how you would calculate the expected round length. When mining Bitcoins, round length percentiles (the chances a round will go as long as it did) are a binomial distribution based on the number of total independent hashes. What you are doing is equivalent to taking interest compounded monthly, using stats about the interest where it is compounded yearly, and trying to fit it to a curve that would be for interest compounded continuously. Actually, it looks like you are just adding up the total number of shares, instead of fitting them to a round quantile, which is even farther from what you should be doing.

Here is an example of how percentile calculations using just the number of shares and ignoring the underlying difficulty 1 binomial distribution of share finding will skew your results. This is using real binomial percentile calculation instead of whatever you are doing...
This is a table of all block finds for the last difficulty at BTCg.

#   Block   Date/Time   Duration   Shares   Total Hashes percentile   share percentile
2362   141124   08/16/11 12:11 AM   04:56:00   7609114   02.656%   01.479%
2363   141125   08/16/11 12:37 AM   00:26:00   0681963   72.239%   68.546%
2364   141130   08/16/11 01:29 AM   00:52:00   1350501   52.520%   47.335%
2365   141135   08/16/11 02:25 AM   00:56:25   1484443   49.271%   43.951%
2366   141136   08/16/11 02:31 AM   00:05:30   0143481   93.387%   92.361%
2367   141147   08/16/11 05:31 AM   03:00:05   4832611   09.982%   06.882%
2368   141153   08/16/11 06:09 AM   00:38:01   1054539   60.481%   55.766%
2369   141160   08/16/11 07:12 AM   01:03:00   1735924   43.703%   38.237%
2370   141162   08/16/11 07:36 AM   00:24:31   0698169   71.683%   67.933%
2371   141165   08/16/11 07:56 AM   00:19:28   0529012   77.705%   74.605%
2372   141168   08/16/11 08:44 AM   00:48:00   1358208   52.328%   47.134%
2373   141170   08/16/11 09:07 AM   00:23:14   0667927   72.724%   69.080%
2374   141173   08/16/11 10:16 AM   01:08:56   1926729   39.902%   34.403%
2375   141179   08/16/11 11:05 AM   00:49:23   1385716   51.646%   46.421%
2376   141182   08/16/11 11:41 AM   00:35:28   0972989   62.879%   58.342%
2377   141183   08/16/11 11:46 AM   00:04:59   0135719   93.733%   92.759%
2378   141188   08/16/11 12:21 PM   00:35:54   1012043   61.719%   57.094%
2379   141198   08/16/11 01:41 PM   01:19:07   2209830   34.864%   29.411%
2380   141203   08/16/11 02:05 PM   00:23:59   0667575   72.737%   69.094%
2381   141209   08/16/11 02:38 PM   00:33:11   0942983   63.785%   59.320%
2382   141211   08/16/11 02:45 PM   00:06:49   0183003   91.644%   90.362%
2383   141215   08/16/11 03:10 PM   00:25:02   0700124   71.616%   67.860%
2384   141218   08/16/11 03:32 PM   00:21:58   0607330   74.856%   71.438%
2385   141220   08/16/11 03:53 PM   00:21:39   0616836   74.518%   71.063%
2386   141223   08/16/11 04:19 PM   00:25:21   0696526   71.739%   67.995%
2387   141230   08/16/11 05:49 PM   01:30:01   2501908   30.331%   25.018%
2388   141242   08/16/11 08:00 PM   02:11:00   3533660   18.545%   14.129%
2389   141245   08/16/11 08:21 PM   00:21:25   0578235   75.902%   72.598%
2390   141247   08/16/11 08:48 PM   00:27:14   0677616   72.389%   68.711%
2391   141252   08/16/11 09:42 PM   00:53:20   1266941   54.655%   49.577%
2392   141256   08/16/11 10:06 PM   00:24:05   0575149   76.014%   72.723%
2393   141266   08/16/11 11:48 PM   01:42:11   2462278   30.910%   25.573%
2394   141274   08/17/11 12:38 AM   00:49:44   1166794   57.329%   52.405%
2395   141277   08/17/11 01:25 AM   00:47:00   0946058   63.692%   59.219%
2396   141278   08/17/11 01:32 AM   00:07:01   0105215   95.107%   94.340%
2397   141279   08/17/11 01:42 AM   00:10:00   0206608   90.618%   89.188%
2398   141283   08/17/11 02:30 AM   00:47:59   1016167   61.598%   56.964%
2399   141295   08/17/11 04:55 AM   02:25:01   3412584   19.647%   15.109%
2400   141306   08/17/11 07:21 AM   02:26:52   3623270   17.769%   13.445%
2401   141308   08/17/11 07:29 AM   00:07:08   0173062   92.079%   90.861%
2402   141310   08/17/11 07:39 AM   00:10:30   0258664   88.396%   86.654%
2403   141326   08/17/11 11:04 AM   03:24:44   5139667   08.623%   05.806%
2404   141331   08/17/11 12:00 PM   00:56:01   1413260   50.972%   45.719%
2405   141333   08/17/11 12:52 PM   00:52:33   1350274   52.526%   47.341%
2406   141341   08/17/11 02:27 PM   01:34:12   2381984   32.116%   26.736%
2407   141342   08/17/11 02:33 PM   00:06:00   0145661   93.290%   92.250%
2408   141358   08/17/11 06:12 PM   03:39:04   5684853   06.649%   04.293%
2409   141380   08/17/11 09:04 PM   02:52:42   4480611   11.806%   08.363%
2410   141382   08/17/11 09:11 PM   00:07:00   0170823   92.177%   90.973%
2411   141384   08/17/11 09:40 PM   00:29:01   0756007   69.733%   65.792%
2412   141389   08/17/11 10:11 PM   00:30:46   0794891   68.452%   64.390%
2413   141392   08/17/11 10:39 PM   00:28:07   0727708   70.681%   66.831%
2414   141401   08/17/11 11:41 PM   01:01:20   1562514   47.470%   42.092%
2415   141411   08/18/11 01:18 AM   01:37:00   2482370   30.615%   25.291%
2416   141419   08/18/11 02:21 AM   01:02:55   1661534   45.281%   39.846%
2417   141430   08/18/11 03:28 AM   01:07:04   1768458   43.030%   37.555%
2418   141431   08/18/11 03:31 AM   00:03:01   0074000   96.533%   95.985%
2419   141445   08/18/11 03:31 AM   00:00:02   0001780   99.915%   99.901%
2420   141478   08/18/11 05:53 AM   02:21:57   3883147   15.698%   11.643%
2421   141450   08/18/11 06:53 AM   01:00:01   1699965   44.459%   39.006%
2422   141432   08/18/11 08:13 AM   01:19:59   2297726   33.433%   28.013%
2423   141457   08/18/11 09:15 AM   01:02:00   1725020   43.931%   38.469%
2424   141461   08/18/11 10:54 AM   01:39:51   2664088   28.074%   22.869%
2425   141470   08/18/11 12:08 PM   01:13:10   2005153   38.438%   32.941%
2426   141487   08/18/11 01:34 PM   01:25:59   2386828   32.042%   26.665%
2427   141496   08/18/11 03:20 PM   01:46:43   2997992   23.942%   19.008%
2428   141501   08/18/11 04:00 PM   00:39:40   1124282   58.503%   53.653%
2429   141510   08/18/11 05:30 PM   01:29:38   2507835   30.245%   24.936%
2430   141512   08/18/11 05:38 PM   00:08:01   0226044   89.782%   88.233%
2431   141514   08/18/11 05:47 PM   00:08:58   0257777   88.434%   86.696%
2432   141526   08/18/11 07:39 PM   01:52:10   3112459   22.670%   17.841%
2433   141532   08/18/11 08:20 PM   00:40:50   1120369   58.612%   53.770%
2434   141545   08/18/11 10:46 PM   02:26:13   3924076   15.395%   11.382%
2435   141546   08/18/11 11:28 PM   00:42:22   1175575   57.089%   52.151%
2436   141552   08/19/11 01:02 AM   01:33:25   2600589   28.937%   23.688%
2437   141558   08/19/11 01:45 AM   00:43:00   1188897   56.728%   51.767%
2438   141566   08/19/11 03:03 AM   01:18:12   2131499   36.190%   30.715%
2439   141569   08/19/11 03:32 AM   00:28:48   0775002   69.105%   65.103%
2440   141573   08/19/11 04:10 AM   00:38:01   1043501   60.800%   56.108%
2441   141574   08/19/11 04:26 AM   00:16:34   0467272   80.026%   77.200%
2442   141575   08/19/11 04:42 AM   00:15:25   0415757   82.017%   79.434%
2443   141586   08/19/11 06:27 AM   01:45:01   2984100   24.101%   19.155%
2444   141600   08/19/11 09:12 AM   02:45:39   4779202   10.240%   07.088%
2445   141605   08/19/11 09:39 AM   00:26:20   0753706   69.810%   65.875%
2446   141615   08/19/11 11:44 AM   02:05:04   3631839   17.697%   13.381%
2447   141616   08/19/11 12:00 PM   00:15:53   0482024   79.466%   76.571%
2448   141618   08/19/11 12:25 PM   00:25:03   0733702   70.479%   66.609%
2449   141619   08/19/11 12:32 PM   00:07:01   0205693   90.657%   89.234%
2450   141622   08/19/11 01:18 PM   00:46:11   1371842   51.989%   46.779%
2451   141628   08/19/11 01:46 PM   00:27:48   0797782   68.358%   64.287%
2452   141631   08/19/11 02:47 PM   01:01:00   1799350   42.401%   36.918%
2453   141644   08/19/11 04:51 PM   02:04:24   3688955   17.221%   12.965%
2454   141671   08/19/11 10:38 PM   05:46:41   9937498   00.875%   00.407%
2455   141673   08/19/11 10:53 PM   00:14:52   0430853   81.428%   78.772%
2456   141674   08/19/11 11:03 PM   00:10:28   0299371   86.697%   84.722%
2457   141695   08/20/11 02:28 AM   03:25:21   5818787   06.237%   03.986%
2458   141701   08/20/11 03:17 AM   00:48:15   1391035   51.515%   46.285%
2459   141703   08/20/11 03:54 AM   00:36:59   1074718   59.902%   55.146%
2460   141709   08/20/11 04:29 AM   00:35:01   1003017   61.985%   57.380%
2461   141711   08/20/11 04:44 AM   00:15:10   0449461   80.709%   77.965%
2462   Invalid   08/20/11 06:38 AM   01:53:50   3305441   20.677%   16.032%
2463   141733   08/20/11 08:25 AM   01:47:02   3212302   21.616%   16.881%
2464   141734   08/20/11 08:31 AM   00:06:34   0189122   91.377%   90.056%
2465   141747   08/20/11 10:14 AM   01:42:29   3054329   23.307%   18.424%
2466   141756   08/20/11 11:44 AM   01:30:44   2728916   27.219%   22.063%
2467   141759   08/20/11 11:50 AM   00:05:29   0157588   92.761%   91.643%
2468   141760   08/20/11 12:18 PM   00:27:47   0835741   67.132%   62.950%
2469   141780   08/20/11 02:20 PM   02:02:25   3677391   17.316%   13.048%
2470   141782   08/20/11 03:47 PM   01:26:34   2558765   29.520%   24.243%
2471   141784   08/20/11 03:56 PM   00:09:00   0274134   87.747%   85.915%
2472   141767   08/20/11 04:08 PM   00:12:00   0360555   84.204%   81.900%
2473   141781   08/20/11 04:10 PM   00:02:01   0039201   98.148%   97.852%
2474   141802   08/20/11 07:51 PM   03:41:47   6420919   04.681%   02.856%
2475   141810   08/20/11 09:23 PM   01:31:16   2588165   29.109%   23.851%
2476   141815   08/20/11 09:37 PM   00:13:56   0380546   83.405%   80.998%
2477   141818   08/20/11 09:48 PM   00:11:01   0308384   86.325%   84.300%
2478   141821   08/20/11 10:35 PM   00:47:01   1333068   52.959%   47.795%
2479   141828   08/20/11 11:29 PM   00:53:58   1546878   47.826%   42.458%
2480   141834   08/21/11 12:26 AM   00:57:00   1621428   46.155%   40.740%
2481   141836   08/21/11 12:53 AM   00:27:07   0782568   68.856%   64.831%
2482   141854   08/21/11 04:32 AM   03:38:53   6243885   05.093%   03.150%
2483   141855   08/21/11 04:46 AM   00:14:44   0444035   80.918%   78.199%
2484   141864   08/21/11 06:13 AM   01:26:55   2516246   30.124%   24.820%
2485   141869   08/21/11 06:34 AM   00:20:26   0580633   75.816%   72.502%
2486   141870   08/21/11 06:49 AM   00:14:45   0451507   80.630%   77.877%
2487   141874   08/21/11 07:36 AM   00:47:36   1400203   51.290%   46.050%
2488   141877   08/21/11 07:57 AM   00:21:03   0616930   74.515%   71.059%
2489   141894   08/21/11 10:35 AM   02:37:31   4647383   10.904%   07.625%
2490   141896   08/21/11 10:38 AM   00:02:51   0088585   95.864%   95.213%
2491   141901   08/21/11 11:24 AM   00:46:10   1363241   52.202%   47.003%
2492   141903   08/21/11 12:24 PM   00:59:59   1674707   44.998%   39.556%
2493   141911   08/21/11 01:23 PM   00:59:00   1711755   44.210%   38.753%
2494   141915   08/21/11 01:39 PM   00:16:00   0474834   79.738%   76.877%
2495   141914   08/21/11 01:40 PM   00:00:50   0021798   98.966%   98.800%
2496   141916   08/21/11 01:48 PM   00:08:10   0232932   89.488%   87.898%
2497   141923   08/21/11 03:09 PM   01:21:01   2249399   34.212%   28.773%
2498   141924   08/21/11 03:10 PM   00:01:00   0031841   98.493%   98.252%
2499   141935   08/21/11 05:03 PM   01:53:06   3301770   20.713%   16.065%
2500   141951   08/21/11 07:14 PM   02:11:24   3699692   17.133%   12.888%
2501   141955   08/21/11 07:40 PM   00:25:29   0668345   72.710%   69.064%
2502   141968   08/21/11 09:38 PM   01:58:07   3137588   22.400%   17.594%
2503   141980   08/21/11 11:11 PM   01:32:54   2377221   32.189%   26.807%
2504   141990   08/21/11 11:57 PM   00:46:09   1176870   57.054%   52.113%
2505   141991   08/21/11 11:59 PM   00:01:50   0045022   97.876%   97.538%
2506   141993   08/22/11 12:07 AM   00:08:01   0205596   90.662%   89.238%
2507   142001   08/22/11 12:54 AM   00:47:44   1245173   55.226%   50.179%
2508   142003   08/22/11 01:02 AM   00:07:15   0174270   92.026%   90.800%
2509   142004   08/22/11 01:07 AM   00:05:00   0136568   93.695%   92.716%
2510   142005   08/22/11 01:21 AM   00:14:39   0388225   83.100%   80.654%
2511   142011   08/22/11 02:12 AM   00:50:22   1298670   53.835%   48.714%
2512   142015   08/22/11 02:42 AM   00:30:38   0813555   67.846%   63.728%
2513   142016   08/22/11 02:45 AM   00:02:21   0049561   97.664%   97.293%
2514   142025   08/22/11 04:23 AM   01:38:01   2551980   29.615%   24.334%
2515   142033   08/22/11 05:20 AM   00:57:38   1542625   47.923%   42.558%
2516   142055   08/22/11 07:48 AM   02:27:22   4054779   14.465%   10.587%
2517   142059   08/22/11 08:38 AM   00:50:20   1424492   50.700%   45.435%
2518   142061   08/22/11 09:42 AM   01:03:54   1845962   41.469%   35.977%
2519   142064   08/22/11 09:53 AM   00:10:45   0284980   87.294%   85.400%
2520   142065   08/22/11 10:00 AM   00:07:02   0195652   91.093%   89.731%
2521   142088   08/22/11 12:55 PM   02:54:58   4960234   09.393%   06.412%
2522   142099   08/22/11 03:04 PM   02:09:43   3686401   17.242%   12.983%
2523   142116   08/22/11 05:49 PM   02:44:55   4656867   10.855%   07.585%
2524   142118   08/22/11 06:09 PM   00:19:22   0543406   77.173%   74.012%
2525   142121   08/22/11 06:31 PM   00:22:36   0635583   73.855%   70.329%
2526   142129   08/22/11 08:03 PM   01:31:24   2545096   29.713%   24.427%
2527   142140   08/22/11 09:28 PM   01:25:00   2364648   32.383%   26.994%
2528   142141   08/22/11 09:36 PM   00:08:04   0233339   89.470%   87.878%
2529   142143   08/22/11 09:50 PM   00:14:00   0375147   83.620%   81.240%
2530   142146   08/22/11 10:00 PM   00:09:56   0271610   87.852%   86.035%
2531   142168   08/23/11 12:57 AM   02:57:12   4958392   09.401%   06.419%
2532   142173   08/23/11 01:58 AM   01:01:36   1762323   43.156%   37.682%
2533   142174   08/23/11 02:06 AM   00:07:03   0200533   90.881%   89.489%
2534   142184   08/23/11 02:48 AM   00:42:09   1185587   56.817%   51.862%
2535   142194   08/23/11 04:43 AM   01:54:50   3324331   20.491%   15.866%
2536   142195   08/23/11 04:50 AM   00:07:26   0216254   90.202%   88.713%
2537   142202   08/23/11 05:56 AM   01:05:37   1913737   40.150%   34.651%
2538   142211   08/23/11 07:16 AM   01:20:48   2284651   33.642%   28.217%
2539   142213   08/23/11 07:42 AM   00:25:52   0753785   69.807%   65.873%
2540   142221   08/23/11 08:30 AM   00:47:27   1348127   52.580%   47.398%
2541   142230   08/23/11 09:33 AM   01:03:01   1811406   42.158%   36.672%
2542   142231   08/23/11 09:39 AM   00:06:00   0170895   92.174%   90.970%
2543   142244   08/23/11 10:59 AM   01:20:00   2283738   33.656%   28.231%
2544   142251   08/23/11 12:14 PM   01:15:21   2200739   35.015%   29.559%
2545   142258   08/23/11 01:06 PM   00:51:53   1510217   48.669%   43.328%
2546   142278   08/23/11 03:21 PM   02:15:12   3917862   15.440%   11.421%
2547   142300   08/23/11 08:49 PM   05:28:01   6801826   03.903%   02.312%
2548   142301   08/23/11 08:56 PM   00:06:28   1028646   61.232%   56.571%
2549   142303   08/23/11 09:30 PM   00:34:01   2438248   31.266%   25.916%
2550   142322   08/24/11 12:33 AM   03:03:04   5186095   08.434%   05.658%
2551   142328   08/24/11 02:03 AM   01:29:59   2511933   30.186%   24.880%
2552   142333   08/24/11 02:45 AM   00:42:00   1156510   57.610%   52.704%
2553   142346   08/24/11 05:23 AM   02:38:00   4420084   12.152%   08.648%
2554   142349   08/24/11 05:36 AM   00:13:00   0358914   84.270%   81.974%
2555   142358   08/24/11 07:08 AM   01:32:15   2660622   28.120%   22.913%
2556   142363   08/24/11 07:34 AM   00:26:28   0756036   69.732%   65.791%
2557   142365   08/24/11 07:42 AM   00:08:01   0228131   89.693%   88.132%
2558   142368   08/24/11 07:58 AM   00:15:09   0435106   81.263%   78.587%
2559   142376   08/24/11 09:49 AM   01:50:58   3195791   21.787%   17.036%
2560   142391   08/24/11 11:35 AM   01:46:10   3025469   23.630%   18.721%
2561   142408   08/24/11 02:58 PM   03:22:58   5791868   06.318%   04.046%
2562   142421   08/24/11 06:26 PM   03:28:18   5851416   06.141%   03.914%
2563   142425   08/24/11 08:07 PM   01:40:54   2775808   26.617%   21.497%
2564   142427   08/24/11 08:23 PM   00:15:49   0418619   81.905%   79.308%
2565   142439   08/24/11 09:51 PM   01:28:24   2402909   31.797%   26.428%
2566   142445   08/24/11 11:27 PM   01:35:36   2540231   29.782%   24.493%
2567   142456   08/25/11 01:05 AM   01:38:00   2559087   29.515%   24.239%
2568   142466   08/25/11 02:26 AM   01:21:04   2116696   36.447%   30.968%
2569   142472   08/25/11 04:17 AM   01:50:57   2931173   24.717%   19.725%
2570   142478   08/25/11 04:43 AM   00:26:17   0715194   71.104%   67.296%
2571   142485   08/25/11 05:35 AM   00:51:42   1379627   51.796%   46.578%
2572   142489   08/25/11 06:23 AM   00:48:00   1294650   53.938%   48.822%
2573   142495   08/25/11 07:31 AM   01:08:00   1824487   41.896%   36.407%
2574   142501   08/25/11 08:36 AM   01:05:47   1772265   42.952%   37.475%
2575   142503   08/25/11 08:45 AM   00:08:13   0202233   90.807%   89.405%
2576   142504   08/25/11 08:51 AM   00:06:39   0188678   91.396%   90.078%
2577   142509   08/25/11 09:33 AM   00:41:31   1112043   58.845%   54.018%
2578   142520   08/25/11 11:11 AM   01:37:50   2622165   28.641%   23.406%
2579   142525   08/25/11 12:29 PM   01:18:22   2152234   35.834%   30.364%
2580   142536   08/25/11 03:07 PM   02:37:40   4291742   12.919%   09.285%
2581   142542   08/25/11 05:03 PM   01:56:19   3162430   22.136%   17.354%
2582   142549   08/25/11 06:35 PM   01:31:39   2456122   31.000%   25.661%
2583   142563   08/25/11 09:01 PM   02:26:00   3838146   16.039%   11.936%
2584   142564   08/25/11 09:03 PM   00:02:01   0054250   97.446%   97.040%
2585   142572   08/25/11 10:09 PM   01:05:59   1697227   44.517%   39.066%
2586   142580   08/25/11 11:05 PM   00:56:20   1440643   50.311%   45.030%
2587   142579   08/25/11 11:05 PM   00:00:17   0005253   99.750%   99.710%
2588   142585   08/26/11 12:03 AM   00:57:23   1457687   49.903%   44.607%
2589   142589   08/26/11 12:24 AM   00:21:14   0549533   76.948%   73.762%
2590   142591   08/26/11 12:50 AM   00:25:46   0657659   73.081%   69.474%
2591   142601   08/26/11 01:40 AM   00:50:00   1268784   54.607%   49.527%
2592   142614   08/26/11 03:08 AM   01:28:01   2284627   33.642%   28.217%
2593   142625   08/26/11 05:27 AM   02:18:59   3666314   17.408%   13.128%
2594   142677   08/26/11 05:27 AM   00:00:03   0000663   99.968%   99.963%
2595   142641   08/26/11 06:20 AM   00:52:58   1410142   51.048%   45.798%
2596   142626   08/26/11 07:08 AM   00:47:59   1265316   54.698%   49.622%
2597   142632   08/26/11 01:53 PM   06:45:37   10648774   00.623%   00.275%
2598   142685   08/26/11 04:05 PM   02:11:23   3311627   20.616%   15.978%
2599   142690   08/26/11 04:40 PM   00:35:00   0856172   66.481%   62.241%
2600   142699   08/26/11 06:03 PM   01:22:50   2045320   37.709%   32.216%
2601   142698   08/26/11 06:16 PM   00:13:09   0224481   89.849%   88.310%
2602   142700   08/26/11 06:35 PM   00:19:20   0559432   76.586%   73.358%
2603   142712   08/26/11 08:05 PM   01:30:30   2196947   35.078%   29.621%
2604   142715   08/26/11 08:43 PM   00:37:57   0925473   64.320%   59.898%
2605   142731   08/26/11 11:33 PM   02:49:44   4061634   14.417%   10.547%
2606   142737   08/27/11 12:46 AM   01:12:30   1704244   44.368%   38.914%
2607   142740   08/27/11 01:40 AM   00:54:01   1269611   54.586%   49.504%
2608   142747   08/27/11 04:00 AM   02:19:59   3332412   20.413%   15.795%
2609   142748   08/27/11 04:20 AM   00:19:57   0475795   79.702%   76.836%
2610   142759   08/27/11 05:08 AM   00:48:08   1141242   58.031%   53.152%
2611   142760   08/27/11 05:30 AM   00:22:14   0545689   77.089%   73.919%
2612   142767   08/27/11 06:31 AM   01:00:36   1461931   49.803%   44.503%
2613   142775   08/27/11 08:50 AM   02:19:05   3378500   19.969%   15.397%
2614   142793   08/27/11 11:52 AM   03:02:00   4441617   12.028%   08.545%
2615   142801   08/27/11 01:28 PM   01:36:01   2381076   32.130%   26.750%
2616   142811   08/27/11 03:21 PM   01:52:56   2817633   26.092%   21.005%
2617   142824   08/27/11 04:36 PM   01:15:04   1852772   41.335%   35.841%
2618   142825   08/27/11 04:59 PM   00:22:59   0586722   75.596%   72.258%
2619   142837   08/27/11 07:55 PM   02:56:00   4293643   12.907%   09.275%
2620   142840   08/27/11 08:07 PM   00:12:18   0293147   86.955%   85.015%
2621   142842   08/27/11 08:34 PM   00:26:50   0642278   73.619%   70.069%
2622   142847   08/27/11 10:18 PM   01:43:48   2473242   30.748%   25.419%
2623   142850   08/27/11 11:20 PM   01:02:07   1454249   49.985%   44.692%
2624   142852   08/27/11 11:30 PM   00:10:02   0241919   89.105%   87.461%
2625   142854   08/28/11 12:02 AM   00:32:22   0759580   69.615%   65.662%
2626   142862   08/28/11 02:38 AM   02:35:48   3650511   17.540%   13.244%
2627   142863   08/28/11 02:46 AM   00:07:56   0183507   91.622%   90.337%
2628   142865   08/28/11 03:01 AM   00:15:12   0356376   84.372%   82.089%
2629   142866   08/28/11 03:04 AM   00:02:51   0063392   97.022%   96.550%
2630   142897   08/28/11 08:44 AM   05:39:46   8016526   02.187%   01.180%
2631   142907   08/28/11 11:09 AM   02:24:50   3461476   19.194%   14.705%
2632   142913   08/28/11 12:59 PM   01:50:06   2670659   27.986%   22.786%
2633   142917   08/28/11 01:27 PM   00:27:57   0675141   72.475%   68.805%
2634   142921   08/28/11 02:49 PM   01:22:33   1998135   38.567%   33.069%
2635   142934   08/28/11 04:26 PM   01:36:34   2294743   33.480%   28.060%
2636   142954   08/28/11 08:24 PM   03:58:00   5595685   06.937%   04.510%
2637   142957   08/28/11 08:44 PM   00:20:12   0494324   79.001%   76.052%
2638   142964   08/28/11 10:56 PM   02:11:48   3201088   21.732%   16.986%
2639   142967   08/28/11 11:22 PM   00:26:10   0654882   73.178%   69.581%
2640   142972   08/29/11 12:11 AM   00:49:36   1232654   55.556%   50.528%
2641   142974   08/29/11 12:40 AM   00:28:14   0699448   71.640%   67.885%
2642   142984   08/29/11 01:56 AM   01:16:40   1928985   39.859%   34.360%
2643   142990   08/29/11 02:41 AM   00:44:20   1098627   59.223%   54.421%
2644   142993   08/29/11 03:06 AM   00:25:04   0648936   73.386%   69.811%
2645   142994   08/29/11 03:14 AM   00:08:00   0201389   90.844%   89.446%
2646   142996   08/29/11 03:44 AM   00:30:19   0768774   69.310%   65.328%
2647   143002   08/29/11 05:57 AM   02:12:50   3437147   19.418%   14.905%
2648   143016   08/29/11 09:12 AM   03:14:48   5176607   08.472%   05.688%
2649   143028   08/29/11 10:55 AM   01:43:31   2795485   26.369%   21.264%
2650   143043   08/29/11 12:02 PM   01:06:57   1796155   42.466%   36.983%
2651   143048   08/29/11 12:40 PM   00:37:39   1030350   61.182%   56.518%
2652   143052   08/29/11 01:36 PM   00:56:22   1550462   47.744%   42.373%
2653   143062   08/29/11 04:16 PM   02:39:51   4299717   12.870%   09.244%
2654   143074   08/29/11 06:44 PM   02:27:59   3974780   15.027%   11.067%
2655   143087   08/29/11 09:11 PM   02:27:11   3895349   15.607%   11.564%
2656   143091   08/29/11 10:14 PM   01:02:23   1644946   45.641%   40.213%
2657   143092   08/29/11 10:23 PM   00:09:27   0261903   88.260%   86.499%
2658   143093   08/29/11 10:30 PM   00:06:59   0184834   91.564%   90.270%
2659   143110   08/30/11 01:08 AM   02:38:13   4194390   13.533%   09.799%
2660   143111   08/30/11 01:17 AM   00:08:21   0214580   90.274%   88.795%
2661   143114   08/30/11 01:41 AM   00:24:31   0650079   73.346%   69.767%
2662   143115   08/30/11 01:50 AM   00:09:10   0242691   89.072%   87.424%
2663   143120   08/30/11 02:48 AM   00:57:39   1536122   48.072%   42.711%
2664   143127   08/30/11 03:58 AM   01:10:14   1872587   40.946%   35.450%
Mean percentile using estimated total hashes: 52.41%
Mean percentile using shares vs difficulty: 48.60%
Mean # of shares:1831414.1320132      

(We can see above that accusations of the pool being unlucky are already specious, as the pool found blocks with 101.4% of the expected shares, pretty darn close.)

As you can see, the last two percentile columns do not match. in the first, I have estimated the number of shares per difficulty 1 block solve to get the total hashes computed at that round's difficulty, and then determined the percentile for the full probability; in the second I have naively just used the number of shares. You can see that I get different results. This is the problem with just using somebody's CDF calculator without understanding the statistics behind it.

Secondly, you are making some kind of statement that BTC guild has been improbably unlucky. Could we can blame the luck of the miners? Maybe their submitted shares are unlucky? We must first analyze if we can expect the difficulty 1 block solves to correlate to hash rate.

For this I use the R statistics package. The probability of a single hash producing a block solve at difficulty 1 is 2^-32, and I am going to find the 95% confidence interval of probability for a single block find:

R version 2.13.1 (2011-07-08)
Copyright (C) 2011 The R Foundation for Statistical Computing

> dif1prob <- 2^-32
> dif1trials <- 2^32
> oneblock <- 1
> binom.test(x = oneblock, n = dif1trials, p = dif1prob, conf.level = 0.95)

        Exact binomial test

data:  oneblock and dif1trials 
number of successes = 1, number of trials = 4294967296, p-value = 1
alternative hypothesis: true probability of success is not equal to 2.328306e-10 
95 percent confidence interval:
 5.894762e-12 1.297249e-09 
sample estimates:
probability of success 
          2.328306e-10 

Here we can see that if we find a single block at difficulty 1 with the expected average number of hashes, we would have a very wide number of hashes possible for a block solve where we can still say with 95% confidence that the probability is correct. We also know intuitively that a lucky person could find a block in just a few hashes, which would lie outside the confidence interval, showing that even statistics can't absolutely doubt random events. How about if we increase this to 100,000 blocks at the same block solve rate?:

> binom.test(x = oneblock*100000, n = dif1trials*100000, p = dif1prob, conf.level = 0.95)

        Exact binomial test

data:  oneblock * 1e+05 and dif1trials * 1e+05 
number of successes = 1e+05, number of trials = 4.294967e+14, p-value = 1
alternative hypothesis: true probability of success is not equal to 2.328306e-10 
95 percent confidence interval:
 2.313898e-10 2.342786e-10 
sample estimates:
probability of success 
          2.328306e-10 

Here we can see that the confidence interval lies between 2.31 and 2.34 (x 10^-10), a much smaller range, so we can intuit that with over 100,000 shares submitted, with 95% confidence that the luck of share finding shouldn't deviate more than 1% (but in 1 of 20 cases we can predict it will).

So now we look at the blocks of BTC guild and see if we have enough statistics to make any conclusion. Let's pick a rather "unlucky" day, 8/24, where 17 blocks were found with 43715542 shares (an average of 2571502 shares per round, at difficulty 1805700.83). I'm going to compute with just shares first so I don't run out of memory...:

> x<-17
> N<-43715542
> prob<-(1/1805700.83)
> dbinom(x, N, prob)
[1] 0.02901398

Here we get the percentile of 2.9%; these odds can be expected averaging 1 in 34 days.
Can we say though that the game is "fixed", that the probability this day is not what is expected?

> binom.test(x, N, prob)

        Exact binomial test

data:  x and N 
number of successes = 17, number of trials = 43715542, p-value = 0.1547
alternative hypothesis: true probability of success is not equal to 5.538016e-07 
95 percent confidence interval:
 2.265356e-07 6.226308e-07 
sample estimates:
probability of success 
          3.888777e-07

The "alternative hypothesis" as represented above is what would be outside the confidence level, that the probability is not the Bitcoin probability, maybe luck being affected by the pool op. We can see that both the bitcoin probability and the probability estimate from the results are both within the confidence interval, meaning that these results do not deviate outside an expected block solve rate for the day.

These numbers are now large enough that the discrete binomial distribution has approached the continuous poisson model for the same data (shown below), and we can then do even bigger math:

> poisson.test(x, N, prob)

        Exact Poisson test

data:  x time base: N 
number of events = 17, time base = 43715542, p-value = 0.1547
alternative hypothesis: true event rate is not equal to 5.538016e-07 
95 percent confidence interval:
 2.265356e-07 6.226309e-07 
sample estimates:
  event rate 
3.888777e-07 

Now let's look at the data for the whole two-week difficulty period where 303 blocks were found with 554918482 shares:

> x2w <-303
> N2w <-554918482
> poisson.test(x2w, N2w, prob)

        Exact Poisson test

data:  x2w time base: N2w 
number of events = 303, time base = 554918482, p-value = 0.8417
alternative hypothesis: true event rate is not equal to 5.538016e-07 
95 percent confidence interval:
 4.862695e-07 6.110991e-07 
sample estimates:
  event rate 
5.460261e-07

Lets do the full expansion based on the total hashrate, and even go a step further and say that the shares being submitted this whole time were 0.5% luckier than expected, resulting in more hashrate trials per submitted share:

> poisson.test(x2w, ((N2w*2^32)*1.005), (prob*2^-32))

        Exact Poisson test

data:  x2w time base: ((N2w * 2^32) * 1.005) 
number of events = 303, time base = 2.395274e+18, p-value = 0.776
alternative hypothesis: true event rate is not equal to 1.28942e-16 
95 percent confidence interval:
 1.126552e-16 1.415747e-16 
sample estimates:
  event rate 
1.264991e-16

I think you get the idea, the estimated block win rate of BTC Guild is well within the expected confidence range, the Vladimir hypothesis is busted.

Edit: maybe clearer, probably not...