Since this is at least nominally a probability blog, I draw your attention to the following: Call p and p+2 a twin prime pair if they're both prime. Then it's conjectured that the number of twin prime pairs with smaller member less than or equal to x, denoted π2(x), is asymptotically
and there is probabilistic reasoning that leads to this, which is a version of the Twin Prime Conjecture; basically, the "probability" that a number near t is prime is 1/(log t), and so the probability that two numbers near t are both prime should be the square of this. So the number of twin prime pairs under x should be just that integral, except that we have to take some divisiblity information into account, which is what the product out front does. If a number n is odd, then n+2 is twice as likely to be prime as it would ``otherwise" be; if n is not divisible by some odd prime p, then n+2 has ``probability" (p-2)/(p-1) of being not divisible by p (we must have that n isn't two less than a multiple of p), which is 1 - 1/(p-1)2 times the "naive" probability that n+2 isn't divisible by p, namely (p-1)/p.
I rather like results of this form, where we can guess properties of the primes by assuming that the probability that a given integer is divisible by a prime p is 1/p and these events are independent for different primes p; I've talked about this before, when I sketched a heuristic argument for the Goldbach conjecture. Sure,.the primes have no business behaving randomly. But what reason is there that they should behave nonrandomly?