20160307, 23:36  #1 
"Ron"
Jan 2016
Fitchburg, MA
97 Posts 
Odds
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)? Last fiddled with by Fred on 20160307 at 23:42 
20160307, 23:45  #2 
"Curtis"
Feb 2005
Riverside, CA
5^{3}×37 Posts 
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. 
20160308, 01:54  #3 
"Ron"
Jan 2016
Fitchburg, MA
97 Posts 
Thanks! Great answer and very helpful.

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