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Message boards : Project Staging Area : Chance of ESP Prime

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Profile Roger
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Found 1 prime in the 2018 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (2,037,982)Cullen LLR Ruby: Earned 2,000,000 credits (2,015,907)ESP LLR Ruby: Earned 2,000,000 credits (2,232,391)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,088,705)PPS LLR Ruby: Earned 2,000,000 credits (2,962,062)PSP LLR Ruby: Earned 2,000,000 credits (2,539,644)SoB LLR Ruby: Earned 2,000,000 credits (2,122,524)SR5 LLR Ruby: Earned 2,000,000 credits (2,238,295)SGS LLR Ruby: Earned 2,000,000 credits (3,619,586)TRP LLR Ruby: Earned 2,000,000 credits (2,125,391)Woodall LLR Ruby: Earned 2,000,000 credits (2,037,732)321 Sieve Turquoise: Earned 5,000,000 credits (5,190,731)Cullen/Woodall Sieve (suspended) Silver: Earned 100,000 credits (207,387)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,049,697)PPS Sieve Double Bronze: Earned 100,000,000 credits (100,422,123)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Ruby: Earned 2,000,000 credits (3,227,972)TRP Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,021,659)AP 26/27 Sapphire: Earned 20,000,000 credits (20,651,644)GFN Emerald: Earned 50,000,000 credits (57,918,585)PSA Sapphire: Earned 20,000,000 credits (43,298,465)
Message 76067 - Posted: 1 May 2014 | 15:02:24 UTC

Chance of ESP Prime Table

+---------------+-------+-------+-------+-------+-------+-------+ | k remaining | 5-6M | 6-7M | 7-8M | 8-9M | 9-10M | 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%| +---------------+-------+-------+-------+-------+-------+-------+

____________

stream
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Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de Primes321 LLR Jade: Earned 10,000,000 credits (10,011,570)Cullen LLR Jade: Earned 10,000,000 credits (10,009,374)ESP LLR Jade: Earned 10,000,000 credits (10,009,221)Generalized Cullen/Woodall LLR Jade: Earned 10,000,000 credits (10,012,217)PPS LLR Jade: Earned 10,000,000 credits (11,055,307)PSP LLR Jade: Earned 10,000,000 credits (10,044,081)SoB LLR Turquoise: Earned 5,000,000 credits (9,993,121)SR5 LLR Turquoise: Earned 5,000,000 credits (9,959,731)SGS LLR Turquoise: Earned 5,000,000 credits (8,929,496)TRP LLR Turquoise: Earned 5,000,000 credits (9,911,706)Woodall LLR Turquoise: Earned 5,000,000 credits (9,963,318)321 Sieve Sapphire: Earned 20,000,000 credits (20,004,228)Generalized Cullen/Woodall Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,047,667)PPS Sieve Sapphire: Earned 20,000,000 credits (20,866,490)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,043,271)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,015,177)AP 26/27 Sapphire: Earned 20,000,000 credits (20,045,194)GFN Emerald: Earned 50,000,000 credits (53,545,357)PSA Double Silver: Earned 200,000,000 credits (200,301,443)
Message 76905 - Posted: 4 Jun 2014 | 10:19:05 UTC - in response to Message 76067.

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.

Profile Toshio Yamaguchi
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321 LLR Bronze: Earned 10,000 credits (16,424)PPS LLR Bronze: Earned 10,000 credits (10,232)SR5 LLR Bronze: Earned 10,000 credits (10,448)TRP LLR Bronze: Earned 10,000 credits (24,275)Woodall LLR Bronze: Earned 10,000 credits (19,632)PPS Sieve Bronze: Earned 10,000 credits (89,027)TRP Sieve (suspended) Bronze: Earned 10,000 credits (20,665)PSA Silver: Earned 100,000 credits (469,144)
Message 76910 - Posted: 4 Jun 2014 | 11:55:05 UTC

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 5-10M range.

stream
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Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de Primes321 LLR Jade: Earned 10,000,000 credits (10,011,570)Cullen LLR Jade: Earned 10,000,000 credits (10,009,374)ESP LLR Jade: Earned 10,000,000 credits (10,009,221)Generalized Cullen/Woodall LLR Jade: Earned 10,000,000 credits (10,012,217)PPS LLR Jade: Earned 10,000,000 credits (11,055,307)PSP LLR Jade: Earned 10,000,000 credits (10,044,081)SoB LLR Turquoise: Earned 5,000,000 credits (9,993,121)SR5 LLR Turquoise: Earned 5,000,000 credits (9,959,731)SGS LLR Turquoise: Earned 5,000,000 credits (8,929,496)TRP LLR Turquoise: Earned 5,000,000 credits (9,911,706)Woodall LLR Turquoise: Earned 5,000,000 credits (9,963,318)321 Sieve Sapphire: Earned 20,000,000 credits (20,004,228)Generalized Cullen/Woodall Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,047,667)PPS Sieve Sapphire: Earned 20,000,000 credits (20,866,490)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,043,271)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,015,177)AP 26/27 Sapphire: Earned 20,000,000 credits (20,045,194)GFN Emerald: Earned 50,000,000 credits (53,545,357)PSA Double Silver: Earned 200,000,000 credits (200,301,443)
Message 76912 - Posted: 4 Jun 2014 | 13:20:38 UTC - in response to Message 76910.

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 | 5-6M | 6-7M | 7-8M | 8-9M | 9-10M | 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 (5-10M).
The "Total" number in the bottom is a probability to find any prime in given range (any k)
And the biggest bottom-right number is a probability that we'll find at least one prime during whole project. It's almost guaranteed!

Profile DaveProject donor
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Credit: 933,798,075
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Found 2 primes in the 2018 Tour de PrimesFound 1 prime in the 2020 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,042,688)Cullen LLR Turquoise: Earned 5,000,000 credits (5,101,836)ESP LLR Turquoise: Earned 5,000,000 credits (5,303,526)Generalized Cullen/Woodall LLR Turquoise: Earned 5,000,000 credits (5,202,873)PPS LLR Turquoise: Earned 5,000,000 credits (6,098,851)PSP LLR Turquoise: Earned 5,000,000 credits (5,513,728)SoB LLR Turquoise: Earned 5,000,000 credits (8,851,996)SR5 LLR Turquoise: Earned 5,000,000 credits (5,199,189)SGS LLR Turquoise: Earned 5,000,000 credits (5,100,018)TRP LLR Turquoise: Earned 5,000,000 credits (5,405,027)Woodall LLR Turquoise: Earned 5,000,000 credits (5,124,820)321 Sieve Jade: Earned 10,000,000 credits (10,003,334)Cullen/Woodall Sieve (suspended) Silver: Earned 100,000 credits (268,250)Generalized Cullen/Woodall Sieve (suspended) Jade: Earned 10,000,000 credits (10,000,502)PPS Sieve Double Silver: Earned 200,000,000 credits (306,761,000)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,000,133)TRP Sieve (suspended) Jade: Earned 10,000,000 credits (10,000,970)AP 26/27 Double Bronze: Earned 100,000,000 credits (190,485,945)GFN Double Bronze: Earned 100,000,000 credits (133,333,389)PSA Double Silver: Earned 200,000,000 credits (200,000,001)
Message 76917 - Posted: 4 Jun 2014 | 16:46:44 UTC

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

Profile Michael GoetzProject donor
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The "Shut up already!" badge:  This loud mouth has mansplained on the forums over 10 thousand times!  Sheesh!!!Discovered the World's First GFN-19 prime!!!Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de Primes321 LLR Ruby: Earned 2,000,000 credits (2,822,730)Cullen LLR Ruby: Earned 2,000,000 credits (2,005,249)ESP LLR Turquoise: Earned 5,000,000 credits (5,009,577)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,145,754)PPS LLR Jade: Earned 10,000,000 credits (15,524,174)PSP LLR Turquoise: Earned 5,000,000 credits (5,197,957)SoB LLR Sapphire: Earned 20,000,000 credits (34,291,181)SR5 LLR Jade: Earned 10,000,000 credits (10,007,110)SGS LLR Ruby: Earned 2,000,000 credits (2,276,330)TRP LLR Turquoise: Earned 5,000,000 credits (5,084,329)Woodall LLR Ruby: Earned 2,000,000 credits (2,195,123)321 Sieve Jade: Earned 10,000,000 credits (10,061,196)Cullen/Woodall Sieve (suspended) Ruby: Earned 2,000,000 credits (4,170,256)Generalized Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,059,304)PPS Sieve Sapphire: Earned 20,000,000 credits (22,885,121)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Amethyst: Earned 1,000,000 credits (1,035,522)TRP Sieve (suspended) Ruby: Earned 2,000,000 credits (2,051,121)AP 26/27 Jade: Earned 10,000,000 credits (10,118,303)GFN Emerald: Earned 50,000,000 credits (74,615,195)PSA Jade: Earned 10,000,000 credits (12,445,029)
Message 76919 - Posted: 4 Jun 2014 | 17:35:51 UTC

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.)


____________
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axn
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321 LLR Gold: Earned 500,000 credits (656,177)Cullen LLR Silver: Earned 100,000 credits (457,297)ESP LLR Silver: Earned 100,000 credits (437,502)PPS LLR Amethyst: Earned 1,000,000 credits (1,256,334)SoB LLR Amethyst: Earned 1,000,000 credits (1,246,109)SR5 LLR Ruby: Earned 2,000,000 credits (2,193,129)SGS LLR Silver: Earned 100,000 credits (101,658)Woodall LLR Silver: Earned 100,000 credits (292,129)Cullen/Woodall Sieve (suspended) Bronze: Earned 10,000 credits (32,498)PPS Sieve Jade: Earned 10,000,000 credits (15,515,229)AP 26/27 Amethyst: Earned 1,000,000 credits (1,491,867)GFN Ruby: Earned 2,000,000 credits (4,245,870)PSA Silver: Earned 100,000 credits (101,309)
Message 76925 - Posted: 5 Jun 2014 | 6:25:35 UTC - in response to Message 76912.
Last modified: 5 Jun 2014 | 6:26:59 UTC

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 | 5-6M | 6-7M | 7-8M | 8-9M | 9-10M | 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 (5-10M).
The "Total" number in the bottom is a probability to find any prime in given range (any k)
And the biggest bottom-right 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.

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Discovered 1 mega primeFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2019 Tour de PrimesFound 1 prime in the 2020 Tour de Primes321 LLR Jade: Earned 10,000,000 credits (10,011,570)Cullen LLR Jade: Earned 10,000,000 credits (10,009,374)ESP LLR Jade: Earned 10,000,000 credits (10,009,221)Generalized Cullen/Woodall LLR Jade: Earned 10,000,000 credits (10,012,217)PPS LLR Jade: Earned 10,000,000 credits (11,055,307)PSP LLR Jade: Earned 10,000,000 credits (10,044,081)SoB LLR Turquoise: Earned 5,000,000 credits (9,993,121)SR5 LLR Turquoise: Earned 5,000,000 credits (9,959,731)SGS LLR Turquoise: Earned 5,000,000 credits (8,929,496)TRP LLR Turquoise: Earned 5,000,000 credits (9,911,706)Woodall LLR Turquoise: Earned 5,000,000 credits (9,963,318)321 Sieve Sapphire: Earned 20,000,000 credits (20,004,228)Generalized Cullen/Woodall Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,047,667)PPS Sieve Sapphire: Earned 20,000,000 credits (20,866,490)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,043,271)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,015,177)AP 26/27 Sapphire: Earned 20,000,000 credits (20,045,194)GFN Emerald: Earned 50,000,000 credits (53,545,357)PSA Double Silver: Earned 200,000,000 credits (200,301,443)
Message 76927 - Posted: 5 Jun 2014 | 7:35:04 UTC - in response to Message 76925.

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.

Message boards : Project Staging Area : Chance of ESP Prime

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