Imagine that I need to verify that a specific integer, like $n = 10^{1347}+3049$ is prime.

`ProvablePrimeQ` takes a lot of time, and, even if it will eventually work for this integer, it will work forever for somewhat larger integers.

`PrimeQ` works in 0.156 seconds, but, according to specification, it uses "the multiple Rabin-Miller test in bases 2 and 3 combined with a Lucas pseudoprime test". As far as I understand, its answer for every specific number is deterministic - either correct or false (no probabilities involved). It may happen that it returns false answer for $n=10^{1347}+3049$, and, in this case, it will still return false answer even if I call `PrimeQ[n]` a million times.

I do not understand why use Rabin-Miller test in bases 2 and 3 only? The theory says that if we select bases AT RANDOM, we get the correct answer with probability $3/4$. We can then repeat the test $N$ times to reduce the probability of error down to $(1/4)^N$. This way I can control the error probability. 

The question is: Is there a randomised primality test implemented in Mathematica, where I can control the error probability as a parameter? Or where the error probability is given in the specification and I can reduce it by repeating the test?