While playing with programs to determine primes and relative
primes, I stumbled over an interesting (at least to me) fact. While the
probability of a random number being prime decreases as the range of possible
random numbers increases (Prime Number Theorem), the probability of two
random numbers being relatively prime is 60.8% Is this something that
is either well known by or trivially obvious to prime number gurus?
That there should be such a constant is "obvious", finding its value takes
more work. In 1849 Dirichlet showed that the probability is 6/2
roughly as follows.
Suppose you pick two random numbers less than n, then
[n/2]2 pairs are both divisible by 2.
[n/3]2 pairs are both divisible by 3.
[n/5]2 pairs are both divisible by 5.
...
(Here [x] is the greatest integer less than or equal to x,
usually called the floor function.) So the number of relatively prime
pairs less than or equal to n is (by the inclusion/exclusion principle):
where the sums are taken over the primes p,q,r,... less than n.
Letting mu(x) be the möbius function this is
sum(mu(k)[n/k]2)
(sum over positive integers k)
so the desired constant is the limit as n goes to infinity of this
sum divided by n2, or
sum(mu(k)/k2) (sum
over positive integers k).
But this series times the sum of the reciprocals of the squares is one,
so the sum of this series, the desired limit, is 6/2.
This number is approximately
2
roughly as follows. 
