Exactly. You get a pair every 17 hands, which is another way of saying
don't wait for the bus! But how often do they show up at the table, in
anybody's hand?
As for the odds of a pair in the community cards, that is simply the
odds of a pair in any 3 random cards, so to get a sense for it you
could shuffle a pack of cards and take off three-card packets
one by one. By the calculation below, you should get a pair almost
1 out of every 5 times (specifically, 17%).
Code:
Number of possible 3-card combinations:
n = 52 * 51 * 50 = 132600
Number of possible 3-card combinations _without_ a pair:
m = 52 * 48 * 44 = 109824
Probability of 3-card combination not having a pair:
m / n = 0.8282
So probability of 3-card combination having a pair:
1 - m / n = 0.1717647