Odds

[Fred]

Could someone give a brief elementary level explanation of how Prime95 calculates the odds that one of the numbers someone is testing is prime? For example, under Test -> Status where it says the odds are 1 out of xxx,xxx.

I had thought the number of Mersenne primes was thought (proven) to be infinite. Is there a way of calculating (or estimating) how many there are in a given exponent range? Or perhaps are the odds give just based on past experience (number of exponents tested by gimps in relation to number of primes found by gimps)?

[VBCurtis]

The most elementary answer: http://primes.utm.edu/howmany.html
The Prime Number Theorem. Consequence 3 specifically.

However, Prime95 takes into account the trial factoring effort that has been done; a candidate Mersenne that has passed TF to, say, 70 bit is much more likely to be prime than a candidate that nothing is known about. For that, you need a more complicated formula (one also presented elsewhere in this forum- I refer you to the search function, now that you have "prime number theorem" or "odds of prime" search terms).

Your question 2 is answered by the first solution- if one can calculate the chances each test comes back prime, one can easily find expected number of primes in a set of 1000 or 10000 (etc) tests. If a single test has 1/50000 chance to be prime, and I run 100000 such tests, I can expect two primes. Note that is NOT the odds of finding a prime in such a range- expectation and probability are separate but related calculations.

[Fred]

Thanks! Great answer and very helpful.

[axn]

The Math

[kladner]

Originally Posted by axn

The Math

Thanks! I still don't really understand the math, but I get a broad sense of the operations carried out.
One small step. No giant leaps here!