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Chance of ESP Prime
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RogerVolunteer developer Volunteer tester
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Joined: 27 Nov 11 Posts: 1137 ID: 120786 Credit: 267,001,662 RAC: 7,952

Chance of ESP Prime Table
++++++++
 k remaining  56M  67M  78M  89M  910M  Total 
++++++++
 131179  5.19%  4.38%  3.80%  3.35%  3.00% 19.71% 
 161041  6.57%  5.55%  4.81%  4.24%  3.80% 24.97% 
 163187  3.96%  3.34%  2.90%  2.56%  2.29% 15.04% 
 193997  2.96%  2.50%  2.17%  1.91%  1.71% 11.24% 
 200749  4.70%  3.97%  3.44%  3.03%  2.71% 17.86% 
 202705 12.75% 10.78%  9.34%  8.24%  7.37% 48.46% 
 209611  3.03%  2.56%  2.22%  1.96%  1.75% 11.52% 
 227723  4.72%  3.99%  3.46%  3.05%  2.73% 17.95% 
 229673  2.33%  1.97%  1.71%  1.50%  1.35%  8.86% 
 238411  7.58%  6.41%  5.55%  4.90%  4.38% 28.83% 
 91549  3.43%  2.90%  2.51%  2.21%  1.98% 13.02% 
 99739  5.25%  4.44%  3.85%  3.39%  3.03% 19.96% 
++++++++
 Total  62.45% 52.80% 45.74% 40.35% 36.09%237.44%
++++++++
____________
 

streamVolunteer moderator Project administrator Volunteer developer Volunteer tester Send message
Joined: 1 Mar 14 Posts: 792 ID: 301928 Credit: 484,764,416 RAC: 144,660

237%. You made my day. I could agree that correct formula is long and boring to write, so it could be better not to include these totals at all. But plain summing of probabilities is an ugliest error.
 


I don't immediately see a problem if the probabilities for the different k's are independent of each other and don't think the greater than 100% total percentage is an error in that respect. I interpret a value of over 200% as indicating that two primes can be expected in the 510M range.  

streamVolunteer moderator Project administrator Volunteer developer Volunteer tester Send message
Joined: 1 Mar 14 Posts: 792 ID: 301928 Credit: 484,764,416 RAC: 144,660

Toshio, maximum possible probability is 1, or 100%. There are some formulas how to combine probabilities of events with "or" or "and" conditions, you cannot just add or subtract them.
Ok, with some help from Excel...
++++++++
 k remaining  56M  67M  78M  89M  910M  Total 
++++++++
 91549  3.43%  2.90%  2.51%  2.21%  1.98% 12,37% 
 99739  5.25%  4.44%  3.85%  3.39%  3.03% 18,44% 
 131179  5.19%  4.38%  3.80%  3.35%  3.00% 18,24% 
 161041  6.57%  5.55%  4.81%  4.24%  3.80% 22,62% 
 163187  3.96%  3.34%  2.90%  2.56%  2.29% 14,18% 
 193997  2.96%  2.50%  2.17%  1.91%  1.71% 10,76% 
 200749  4.70%  3.97%  3.44%  3.03%  2.71% 16,63% 
 202705 12.75% 10.78%  9.34%  8.24%  7.37% 40,01% 
 209611  3.03%  2.56%  2.22%  1.96%  1.75% 11,01% 
 227723  4.72%  3.99%  3.46%  3.05%  2.73% 16,72% 
 229673  2.33%  1.97%  1.71%  1.50%  1.35%  8,55% 
 238411  7.58%  6.41%  5.55%  4.90%  4.38% 25,71% 
++++++++
 Total  47,62% 41,92% 37,45% 33,79% 30,80%91,28% 
++++++++
The "Total" number on the right is a probability to find a prime for given k in whole search range (510M).
The "Total" number in the bottom is a probability to find any prime in given range (any k)
And the biggest bottomright number is a probability that we'll find at least one prime during whole project. It's almost guaranteed!
 

Dave Send message
Joined: 13 Feb 12 Posts: 2781 ID: 130544 Credit: 933,798,075 RAC: 368,914

Surely the "totals" should be averages not a sums.  

Michael GoetzVolunteer moderator Project administrator
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Joined: 21 Jan 10 Posts: 13396 ID: 53948 Credit: 228,999,614 RAC: 166,582

Clearly, "the odds of finding a prime" can not exceed 100% or 1.00.
Roger's calculations, however, are not "the odds of finding a prime". Those numbers are "the expected number of primes". That IS a number that can exceed 1.00. (That's assuming we don't stop searching a K after a prime is found, of course.)
That's also why it's appropriate to add the numbers together rather than averaging them.
If, for example, we can expect to find 2 primes while searching K1, and 3 primes while searching K2, then clearly if we search both K1 and K2 we can expect to find 5 primes (not 2.5 primes.)
____________
My lucky number is 75898^{524288}+1  

axnVolunteer developer Send message
Joined: 29 Dec 07 Posts: 285 ID: 16874 Credit: 28,027,106 RAC: 0

Toshio, maximum possible probability is 1, or 100%. There are some formulas how to combine probabilities of events with "or" or "and" conditions, you cannot just add or subtract them.
Ok, with some help from Excel...
++++++++
 k remaining  56M  67M  78M  89M  910M  Total 
++++++++
 91549  3.43%  2.90%  2.51%  2.21%  1.98% 12,37% 
 99739  5.25%  4.44%  3.85%  3.39%  3.03% 18,44% 
 131179  5.19%  4.38%  3.80%  3.35%  3.00% 18,24% 
 161041  6.57%  5.55%  4.81%  4.24%  3.80% 22,62% 
 163187  3.96%  3.34%  2.90%  2.56%  2.29% 14,18% 
 193997  2.96%  2.50%  2.17%  1.91%  1.71% 10,76% 
 200749  4.70%  3.97%  3.44%  3.03%  2.71% 16,63% 
 202705 12.75% 10.78%  9.34%  8.24%  7.37% 40,01% 
 209611  3.03%  2.56%  2.22%  1.96%  1.75% 11,01% 
 227723  4.72%  3.99%  3.46%  3.05%  2.73% 16,72% 
 229673  2.33%  1.97%  1.71%  1.50%  1.35%  8,55% 
 238411  7.58%  6.41%  5.55%  4.90%  4.38% 25,71% 
++++++++
 Total  47,62% 41,92% 37,45% 33,79% 30,80%91,28% 
++++++++
The "Total" number on the right is a probability to find a prime for given k in whole search range (510M).
The "Total" number in the bottom is a probability to find any prime in given range (any k)
And the biggest bottomright number is a probability that we'll find at least one prime during whole project. It's almost guaranteed!
Incidentally, since ESP is a conjecture project where we will stop testing a k as soon as a prime is found, the last column of cumulative probability is also the expected number of primes for the k's. Hence we can sum them up and come up with the expected number of primes for the whole range. It comes to around 2.15 primes. So we can expect to eliminate 2 k's by testing till 10M.  

streamVolunteer moderator Project administrator Volunteer developer Volunteer tester Send message
Joined: 1 Mar 14 Posts: 792 ID: 301928 Credit: 484,764,416 RAC: 144,660

Thanks Michael, I see you point now. Unfortunately, usage of word "chance" and percents in original post was very misleading. Treating these numbers as "density" of primes, final sum, being presented in form of 2.3744, makes sense  meaning that it is expected to be 2 or 3 primes within the whole search range for all k's.  

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