Chance of finding a prime via DC v first LL
 

People seem to be over pessimistic on this question.

The ball park error rate of LLests is 2%.

But DCs are performed on exponents half the size, which
means they take 4 times less GHz-Days, and are twice as
likely to be prime.

So by DCing, your chance of finding a MP per year is 16%
of your chance doing first time LLtests: not as dramatic a
difference as is often portrayed.

David

PS and as Christenson has mentioned elsewhere,
6 x miniscule = miniscule ;-)

[QUOTE=davieddy;265317]People seem to be over pessimistic on this question.

< snip >

But DCs are <snip > are twice as
likely to be prime.

So by DCing, your chance of finding a MP per year is 16%
of your chance doing first time LLtests:[/quote]

The most common error in probability calculations is failing to include the effects and consequences of knowledge.

Your calculation seems to assume that the probability of a DC assigned exponent's being that of a Mersenne prime is just as though it were the corresponding probability for first-time tests -- without using any knowledge gained from the first-time tests. That assumption's not valid!

In actuality, GIMPS [i]does not assign DCs for exponents which are already known to be Mersenne primes![/i]

Thus, a correct calculation would have to take into account that GIMPS is deliberately and knowingly _not_ assigning DCs for exponents of known Mersenne primes.

[quote]not as dramatic a difference as is often portrayed.[/QUOTE]... only because you neglect to account for the effects of prior knowledge!

I deliberately refrained from including the necessary pedantry.

Given that a first time LLtest is erroneous, (2% say), the probability
of the a DC being prime is doubled while the test time is quartered.

What earthly difference does "already found MPs" make to this
sound and simple conditional probability argument?

David

[QUOTE=davieddy;265351]I deliberately refrained from including the necessary pedantry.[/QUOTE]... which allowed misinterpretation, because your actual statement was not your intended statement. :-)

[quote]What earthly difference does "already found MPs" make to this
sound and simple conditional probability argument?[/quote]None, but "this" argument is not the same as the "So by DCing, your chance of finding a MP per year" statement to which I previously replied.

[QUOTE=cheesehead;265354]... which allowed misinterpretation, because your actual statement was not your intended statement. :-)

None, but "this" argument is not the same as the "So by DCing, your chance of finding a MP per year" statement to which I previously replied.[/QUOTE]

The statement stands.

[QUOTE=cheesehead;265349]... only because you neglect to account for the effects of prior knowledge![/QUOTE]

I'm skimming, so I may have missed something important.

But didn't David start with the prior knowledge that only 2% of the first tests are faulty, so if all things were equal, you would be 2% as likely to find a prime.

But these tests, if done correctly, are twice as likely to be prime, so in the same number of tests you are 4% as likely to find a prime.

But the tests are four times faster, so in the same time you are 16% as likely to find a prime.

I don't see what additional prior knowledge would be relevant to the calculation. In particular, knowledge of known Mersenne Primes is not knowledge about faulty tests - those tests are known to be correct, so not relevant to this analysis.

William

[QUOTE=cheesehead;265349]Thus, a correct calculation would have to take into account that GIMPS is deliberately and knowingly _not_ assigning DCs for exponents of known Mersenne primes.[/QUOTE]
My own understanding, and it is likely to be faulty since I have studied no probability theory in more than 25 years, is that the above statement is quite correct in that it identifies a biasing factor. However surely the bias in this case is so tiny (there are so few identified Mersenne Primes) that it will have no effect on the very rough 16% estimate which David arrives at.

[QUOTE=Brian-E;265369]the very rough 16% estimate which David arrives at.[/QUOTE]

THX Brian (and William).

Not all that rough though. OK someone can work out a more accurate
current "error rate" (E%) by the DCs whose residues don't match the
first LL, but I made it clear that "2%" was a ball park guess at E%.

Doubling the exponent increases the GHz-Days by slightly more than 4 times.
The probability of being prime is complicated by "how far TFed" etc, but it involves 1/exponent.

If I said "8E%" instead of "16%" I would not be too far out.

David

Blah blah blah blah blah, you guys should run for office.

Estimating the "error rate"
 

[QUOTE=cheesehead;265349]The most common error in probability calculations is failing to include the effects and consequences of knowledge.

Your calculation seems to assume that the probability of a DC assigned exponent's being that of a Mersenne prime is just as though it were the corresponding probability for first-time tests -- without using any knowledge gained from the first-time tests. That assumption's not valid!

In actuality, GIMPS [I]does not assign DCs for exponents which are already known to be Mersenne primes![/I]

Thus, a correct calculation would have to take into account that GIMPS is deliberately and knowingly _not_ assigning DCs for exponents of known Mersenne primes.

... only because you neglect to account for the effects of prior knowledge![/QUOTE]

I think all your qualms boil down to obtaining an unbiassed estimate
of the probability of the non-zero residue of an unverified LL being wrong.

No doubt a determined quibbler will find this too simple,
but I would take a suitable* large sample of "verified composite via
two matching residues", and find the % where the first test proved
to be wrong.

This totally bypasses your "known prime" "difficulty".

David

* I am "covering my back" this time by allowing
you to decide what "suitable" means!

[QUOTE=LiquidNitrogen;265406]Blah blah blah blah blah, you guys should run for office.[/QUOTE]

More than you needed to know ATM?

"Us guys" (comparative veterans) still enjoy "getting it straight".
:smile:

David