Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$.

Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? Other than the fact that $p \pmod 6$ would have to be $5$? It would seem that the answer has to be no, but I have some perplexing computational results that I would like to account for.