pocket pair getting a set by the river: 12% very close to 1/8
Wrong. PP hitting set on flop is about 12% or 1/8. PP hitting set by river is around 20% or 1/5. You need to know percentages for pot odd purposes and you're making it way too complicated for real time decisions. All you need to do is calculate how many outs you have, multiply that number by 2, and then multiply it by how many cards are yet to come (Flop = 3, Turn = 1, River =1, Turn & River = 2). So with the pocket pair example you used, you have 2 outs to make your set. The odds of hitting it on the flop are 2 outs x 2 x 3 cards coming on flop = 12% (1/8). The odds of making a set by the river are 2 outs x 2 x 5 cards (flop,turn,river) = 20% (1/5). Grant is these number aren't exact, but they're pretty damn close and more than sufficient to calculate pot odds. Do it this way and you can calculate percentages on the fly and it gives you round numbers that are easy to deal with instead of decimals like 11.873645%.