I'm trying to figure two probabilities for a 1 versus 1 texas holdem game. For anyone that is unfamiliar with the game, it is played with a 52 card deck, each player is dealt two cards, and there are community cards dealt out (flop,turn,river).
These community cards are used by each player in combination with their own two cards to make their best possible 5 card hand. The hand values are scored from, high card, to pair, to two pair, three of a kind, straight, flush, full house, 4 of a kind, straight flush, and royal flush. I am only trying to figure out the probability of one of two people making a pair in a head to head game.

The flop is where one card is placed face down ( the burn card) off the deck and 3 are placed face up

The probability of making a pair on each individual flop card is .....
- 12/47 (12 being the 6 possible outs for each player, 47 being 52-4 (2 cards each)-1 (the burn card)
- 12/46 (47 minus the first face up card of the flop)
- 12/45 (46 minus the second face up card of the flop)

the turn is one burn card and one face up.
Its probabilities are...

- 12/43 (45 minus the 3rd face up card of the flop and the burn card)

The river is same as the turn..
Its probabilities are...
- 12/41 (43 minus the face up card of the turn and the burn card)

How do i go about finding the probability of either player pairing on the ENTIRE FLOP, and... pairing on the ENTIRE COMMUNITY? I thought i knew how to add fractions but with probabilities I am apparantly really lost..