TITLE: Probability that a Mersenne number is prime
QUESTION [6 upvotes]: Let $p$ be a prime and let $M_p = 2^p-1$ be a (Mersenne) number. Is there any known result on the probability that $M_p$ is prime? In particular is it known whether the probability tends to $1$ as $p \to \infty$? If so, is there any known lower bound for the asymptotic rate with which this probability tends $1$? Thank you.

REPLY [2 votes]: There was a quotation of the "Wagstaff Mersenne Conjecture", with the claim that for large prime, the probability that $2^p - 1$ is prime is supposed to be about $\frac{\log{p}}{p}$. Since the probability that n is prime is about $\frac{1}{\log{n}}$ this would make the probability that $2^n - 1$ is prime for integer n about $1/n$. The number of Mersenne primes $2^p - 1$ with p ≤ M would be about log M. 
$\log M (2 / \log{2})$, which is about 2.9 times bigger, seems to be a much better estimate for the number of Mersenne primes with exponents ≤ M. It overestimates the number of Mersenne primes by about 3 to 8 in the range where they are known. Some deviation is to be expected. If we picked an odd number near $2^n$ for every n, we would expect the same number of primes. 
So with the known data, $\frac{2\log_2{p}}{p}$ seems to be a better estimate for the probability that $2^p - 1$ is prime if p is a prime.