Flush vs a Full House or Quads draw

[DistantHaze]

I have a question for any math wizards in here: what is the total % chance of a pocket pair beating a flopped flush? For example, if you hold A6s, I hold KK, and the flop is 4s Ks 3s, the odds of a Full House or Quads change depending on the Turn card. For the turn card to pair the board, there are 7/45 cards (3x4, 2xK, 3x3). If the Turn is a 4, for example, then the chance of catching a Boat on the River is 4/44 (2x4, 2xK). If the Turn does NOT pair the board, however, and is an 8 for instance, then the River will make a Full House or Quads 10/44 times (3x3, 3x4, 3x8, 2xK). How do you calculate the COMBINED chance of the pocket Kings beating the Flush?