Calculating odds of two players getting pocket pairs

[AncientPC]

The odds of someone getting a pocket pair is 1:17.

What are the chances of two people getting a pocket pair?

Given that the first pocket pair is TT (i.e. right down the middle), what are the chances of the second pocket pair being higher?

Given 1:7.5 to flop a set, what are the chances that both players with pocket pairs will make a set?

Could you explain the math as well, as I'm having some trouble coming up with these numbers.

[AncientPC]

Someone correct me if my math is wrong.
chance to have pocket pair = 1:16 or 1/17
chance to flop a set with a pocket pair = 1:7.5 or 1/8.5

chance that both players have a pocket pair = 1/17² = 1/289
chance that only one player will flop a set = (1/17 * 1/8.5) * (1/17 * 1-(1/8.5)) = 1/2784
chance that both players will flop a set with a pocket pair = (1/17 * 1/8.5)² = 1/20880

Let's assume you have TT.
chance that someone has a higher pocket pair = 20/52 * 3/51 = 1/44.2
chance that you have TT and someone has a higher pocket pair = 1/17 * (1/17 * 1/44.2) = 1/12774
chances that you have TT, someone has a higher pocket pair, and you both flop sets = (1/17 * 1/8.5) *

((1/17 * 1/44.2) * 1/8.5) = 1/922907 = 0.000108353%

Two things I'd like to point out about the calculations. First of all, the odds for being dealt TT specifically is 1:220 or 1/221 but I used 1/17 for simplicity so that means my numbers are on the high side. Second of all, I realize parentheses aren't required when multiplying fractions but it's for readability.

[garyc8]

The chance that your opponent(assuming you're playing heads up) has a pp in 1/17, on any given hand. The fact that you may have a pp on this hand changes nothing (or very nearly so). If he has a pp, just a hair below 1/3 of them are higher than your TT. (tiny chance he also has TT)
This means that heads up (estimated slightly for the sake of practical, on the spot math) your TT will face a higher pair 1/51.

Of course, in a full game the odds change with each additional player.

As far as losing set over set: If you flop a T it is less than 1/8.5 that your opponent also flops a set. Since one card on flop must be a T, that only leaves him 2 to catch a Q (for example). But of course, with a hand this strong he may well play on and catch on the turn or river.