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we know that the probability that a given $n\in\mathbb{N}$ is a prime is $\frac{1}{\log n}$ and all primes except 2 and 3 are of form $6n\mp 1$. We can deduce that the probability that $6n-1$ is $\frac{1}{2\log n}$? Thank you.

Theory Nombre
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The primes are almost evenly distributed between $6n+1$ and $6n-1$ (and this goes for any such division, like $4n+1$ vs $4n-1$, or $8n+1$ vs $8n+3$ vs $8n+5$ vs $8n+7$). More rigorously, it is known that for a natural number $k$, the ratio between primes below $k$ of one form and primes below $k$ of the other form tends to $1$ as $k$ goes to $\infty$. So as far as probability estimates for large $k$ are concerned, it's relatively safe to assume that they are the same.

However, it has been observed that primes which are quadratic residues (i.e. $6n+1$) do seem to be slightly less common if you look at the difference rather than the ratio. This phenomenon is known as Chebyshev's bias. As pointed out by Joriki in the comments below, this bias has been proven assuming the generalized Riemann hypothesis and the grand simplicity hypothesis, but we don't know yet whether it's actually true.

Arthur
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  • This observation has been proved conditionally on the assumption of the Generalized Riemann Hypothesis and the Grand Simplicity Hypothesis. – joriki Jul 05 '18 at 10:20
  • @joriki Do you mean Chebyshev's bias, or the ratio? – Arthur Jul 05 '18 at 10:44
  • The bias. I was referring to your characterization of it as an "observation" and wanted to point out that it's slightly more than that (though not an unconditionally proven fact). – joriki Jul 05 '18 at 10:47
  • @joriki Cool. I only knew that it was conjectured, not on what grounds. – Arthur Jul 05 '18 at 10:51
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    According to https://en.wikipedia.org/wiki/Chebyshev%27s_bias, I would expect the primes of the form $6k\color{red}{-}1$ to be slightly more common than the primes of the form $6k+1$ (at least for small primes), essentially because the first prime of the form $6k\pm 1$ is $5\equiv -1\pmod{6}$. – Jack D'Aurizio Jul 05 '18 at 14:11
  • @JackD'Aurizio You're right. I read the bias the wrong way. – Arthur Jul 05 '18 at 14:22