20041226, 21:47  #1 
Dec 2003
Paisley Park & Neverland
5×37 Posts 
Error risk after doublecheck
I have some questions regarding the LLerror rate and the safety feature of the res64. Just curious. I'm not good in calculating probabilities...
Here we go:  What's the chance that we have at least one undetected matching but still false doublecheck in the LL database? So both residues are x but should be y.  How many exponents would we have to test so the chance from the question above would rise above 1% ? 
20041226, 23:28  #2 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2×4,597 Posts 
My understanding (and I hope to be corrected if I am wrong) is that the res is a 16 digit Hex number. To get 2 to match accidentally would take odds of (well) over 1 x 10^40, which is the point that it is considered to be impossible. And since the projected starting error rates are ~1%, you need to square that 0.01% chance that the first two res turned in are both wrong.
Ain't going to happen. Unless both were run on thee same error producing machine. 
20041227, 00:07  #3  
May 2004
2^{4}·5 Posts 
Quote:
Dave 

20041227, 00:55  #4 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2×4,597 Posts 
Let's see where (or how) my math breaks down.
16 digits each representing 4 bits. 16 * 4 = 64 bit residue. 2^64 = 18446744073709551616 (total different potential residues) 18446744073709551616^2 = 3.4028e+38 (total different combinations of 2 residues) divide by 0.0001 chance both tests having errors and we get: 3.4028e+42 
20041227, 01:39  #5  
"William"
May 2003
New Haven
2^{3}×5×59 Posts 
Quote:
Alternatively, you can calculate as First error * Second Error * second residue matches first residue 

20041227, 01:57  #6 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
First, I would like to point out that it seems Uncwilly and dave_dm have made an unspoken assumption: that an erroneous residue is equally likely to take on any hexadecimal value. This assumption is necessary to come up with figures at all (unless a more realistic model is available), but it should be kept in mind that there may be some types of CPU errors may not fit this assumption.
In any case, this is how we can come up with the figures. Suppose we actually did two incorrect LL tests of the same exponent. Let x be the residue of the first test. For the second residue, there are 16^16 possible values for the residue (because there are 16 "digits" and each "digit" has 16 possible values). Thus, the probability that both results are the same is: 1/(16^16) ~= 5.42101086242752217003726400434971 * 10^20 Now, we can use a theorem in probability. If the probability that an outcome occurs in one trial is p, then the probability that it doesn't happen is 1  p. If this trial occurs n times, the probability that the outcome never occurs is (1  p)^n. Thus, the probability that it occurs at least once is 1  (1  p)^n. According to http://www.mersenne.org/status.htm, GIMPS has doublechecked a total of 315,179. So for your first question, p = 1/(16^16) and n = 315,179. According to Mathematica, the probability of having at least one such error due to chance is thus roughly: 1.7085887826090294136983459361845392 * 10^14 That's less than one in 58.5 trillion, nothing to get too worried about (if you trust the assumption I pointed out at the beginning). Now, to answer your second question, we solve for n in the equation: 1  (1  p)^n = 0.01 where p = 1/(16^16) (1  p)^n = 0.99 take the log of both sides: n*log(1  p) = log 0.99 n = (log 0.99)/(log (1  p)) Mathematica gives: 1.853959733443683384907575121974264716245215072 * 10^17 Assuming GIMPS doublechecked every possible Mersenne number with a prime exponent in order from (2^2)  1 upwards, in order for the probability of an error to reach 1%, GIMPS would need to search every such exponent up to about: M(7.87 * 10^18) In my opinion, a far greater threat is that a user will deliberately submit incorrect residues to get GIMPS credit. To do this, he could choose any residue he wanted, submit it once manually as a first time LL test, then resubmit it manually as a doublecheck. This could be difficult to catch, especially if he created many user accounts, and never submitted a first time test and its double check using the same account. 
20041227, 02:19  #7 
Oct 2003
Australia, Brisbane
2×5×47 Posts 
Isn't there the little "check sum" or whatever it is at the end of each result line?
I am not exactly sure what it is used for but I always assumed it was there so that people couldn't submit "made up" results. 
20041227, 03:55  #8  
Aug 2002
2^{6}×5 Posts 
Quote:


20041227, 09:29  #9  
May 2004
120_{8} Posts 
Quote:
So with p = 0.01, the probability of two residues matching but some hardware error occuring is q = 2 * p * (1  p) * 2^(64) + p^2 * 2^(64) i.e. {one hardware error} + {two hardware errors} Then log(0.99) / log(1  q) gives the 9.3 x 10^18. Dave 

20041227, 14:19  #10 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2·4,597 Posts 
Ok, so 10^18 is "Ain't gonna happen", while 10^40 is mathametically impossible. :)

20041227, 14:22  #11  
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Quote:


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