I would think that with the great computational skills evident

on this forum that the following derivation would be considered excessive:

Let P

_{p} be the probability that a random prime p is a number n.

By the lemma, P

_{p} = P

_{p-1} = ... .

Hence multiplying the P

_{p} gives

(P

_{p})

^{n} --> 1 or 0

depending on whether there exist any primes.

Additional Lemma: There are primes!

Proof: Start counting at 1 and continue until a number is reached

whose only factors are (well, you know). This process terminates at p=2.

Hence there are primes!

Corollary: The desired probability is 1 (if there really are primes).