  Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH)
and the linear independence hypothesis (LI) on the non-real zeros of
$\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$
has a logarithmic density, which they computed to be about $2.6\times10^{-7}$.
A natural problem is to examine the actual primes in this race. We prove,
assuming RH and LI, that the logarithmic density of the set of primes $p$ for
which $\pi(p)>$ li$(p)$ relative to the prime numbers exists and is the same as
the Rubinstein-Sarnak density. We also extend such results to a broad class of
prime number races, including the "Mertens race" between $\prod_{p<
x}(1-1/p)^{-1}$ and $e^{\gamma}\log x$ and the "Zhang race" between $\sum_{p\ge
x}1/(p\log p)$ and $1/\log x$. These latter results resolve a question of the
first and third author from a previous paper, leading to further progress on a
1988 conjecture of Erd\H{o}s on primitive sets.
