There’s a fun fact that if you choose two numbers at random, the probability that they have no common factors is , or about 61%.
For the two numbers, if you’re looking at any prime p, they’ll have that factor p in common if both their remainders when divided by p are 0.
And if the numbers are truly “random”, their remainders will be evenly distributed and so the odds that they have p is the same as the odds that they both land on 0, which is 1/p².
And it turns out that that infinite product works out to
via math I can’t do. Magic.
But as Wikipedia helpfully points out “There is no way to choose a positive integer at random so that each positive integer occurs with equal probability.”
Intuitively that makes sense. If you pick numbers from a range you’d expect to land “in the middle” on average, but the middle of all integers is unbounded.
I think one of the things that drives that home for me — imagine you spent the rest of your life describing a “RILLY_BIG_NUMBER”. You end up saying “to the power of”, “factorial”, random streams of digits, and stuff like that for the rest of your life. You’re no fun at parties.
At the end you have that RILLY_BIG_NUMBER. But over 99.999…% (say, a RILLY_BIG_NUMBER of 9s) of integers (or primes) are bigger than your number. Because that’s how it is.
