Summary so far: The probability that two uniformly-distributed randomly chosen integers between 1 and n are relatively prime approaches 6/π² for large n. This was established by Ernesto Cesaro in 1885, but nobody seems to refer to it by the name "Cesaro's theorem."

For anyone who's interested, there's a nice explanation here: aperiodical.com/2013/06/cushing-your-luck-properties-of-randomly-chosen-numbers/

I'm out for now, peace.