Existence of a prime number between $x$ and $y$ if $\operatorname{li}(y) - \operatorname{li}(x) = 1$ Is between $x$ and $y$ ($x < y$), there is always at least one prime number (or exactly one?) if $\operatorname{li}(y) - \operatorname{li}(x) = 1$?

Given $x$, the value of $y$ for which li$(y)-{}$li$(x)=1$ is about $y=x+\log x$. It is known that there are infinitely many prime gaps significantly larger than $\log x$, so we know that there are infinitely many intervals $(x,y)$ of this type with no primes.
It is conjectured that the primes follow a Possion-type distribution with parameter $\lambda=1/\log x$; in particular, it is conjectured that intervals of the form $(x,x+\log x)$ contain $k$ primes approximately $e^{-1}/k!$ of the time. In particular, intervals $(x,y)$ of this form should fail to contain primes a positive proportion ($e^{-1}$) of the time.
