If I have a large known prime, is the number 2 greater than that or 2 less than that more likely than chance to be prime?
It seems that since finding new large primes is something of a newsworthy event (especially so if they are the new largest known prime of course), if a twin is more likely than a randomly selected odd number of similar size, once a large prime is found, one could effectively piggyback on that to find other nearby large primes. Even if they are the lower twin, "I found the second largest known prime" would still be of interest to some people, I would think.
It is not known whether there are infinite pairs of twin primes. So, it is possible (although very unlikely) that there is only a finite number of twin primes, in which case for large enough primes $p$ it would be impossible for $p+2$ to be prime (same for $p-2$).
Things get far more complicated in the more plausible scenario that there are infinite pairs of twin prime numbers. We can't say much if we don't know something about the way the twin primes are asymptotically distributed (only that the probability of $p+2$ being prime is positive). And unfortunately we know basically nothing about this.
One famous conjecture of Hardy an d Littlewood -- often regarded as true, but unproven -- establishes that the number of twin number pairs less than $x$ has order
$$ \pi_2(x)\sim C\frac x {(\log x)^2}. $$ for some constant $C$.On the other hand, from the prime number theorem we know that the number of primes less than $x$ is $$ \pi(x)=\frac x {\log x} $$ which is of course of larger order. Hence, if the conjecture is true, then for $p$ odd the ratio of probablilities $$ \frac {P(p+2\ {\rm prime,\ given\ } p\ {\rm prime})} {P(p+2\ {\rm prime})} $$ is asymptotically equal to $$ \left(\frac C {\log x}\right)/\left(\frac 1 {\log x}\right)=C. $$
And, by the way, the constant $C$ has been computed to be approximately $1.32$. So, for very large values of $p$ it will be slightly more likely for $p+2$ to be prime if we know that $p$ is prime. The same story holds for $p-2$.
http://mathworld.wolfram.com/k-TupleConjecture.html