Holding paired cards and flopping EXACTLY two pair by pairing the board.

You are looking for any of the 48 cards that DOES NOT match your hole card. Then you are looking one of the 3 cards that match that card. Then you are looking for a card that DOES NOT match your hole pair and DOES NOT trip up the first two cards of the flop, there are 44 such cards. Then divide by the number of combinations. That gives you:

(48 * 3 * 44 ) / (50 * 49 * 48) = 5.387755%

That is the chance of flopping the board pair on the first two cards. There are two other ways to flop a board pair that have an equal chance: (1) pairing the first and third cards, (2) pairing the second and third cards.

So, you have 3 * 5.387755% = 16.163%

Holding paired cards and flopping EXACTLY trips by flopping a set for your pocket pair.

You are looking for any of the 2 cards that pair your hole cards. Then you are looking for any card that DOES NOT quad up your hole card, there are 48. Then you are looking for a card that DOES NOT quad up your hole card AND DOES NOT match the other flop card (that would give you a full house), there are 44 such cards. Then divide by the number of combinations. That gives you:

(48 * 3 * 44 ) / (50 * 49 * 48) = 3.59184%

That is the chance of flopping the set on the first card. There are two other ways to flop a set that have an equal chance: (1) tripping on the second card, (2) tripping on the third card.

So, you have 3 * 3.59184% = 10.775%

Holding paired cards and flopping EXACTLY a full house, a set to your hole pair and pairing the board.

You are looking for any of the 2 cards that pair your hole cards. Then you are looking for any card that DOES NOT quad up your hole card, there are 48. Then you are looking for a card that pairs up that board card, of which there are 3. Then divide by the number of combinations. That gives you:

(2 * 48 * 3 ) / (50 * 49 * 48) = 0.2448979%

That is the chance of flopping the set on the first card and pairing the second and third. There are two other ways to flop this type of full house that have an equal chance: (1) tripping on the second card, (2) tripping on the third card.

So, you have 3 * 0.2448979% = 0.735% (reviewing my initial post, I transcribed this number incorrectly from my notes. I have fixed the initial post.)

Holding paired cards and flopping EXACTLY a full house, by the board tripping up.

You are looking for any card that does not match your hole pair, there are 48. Then you are looking for any of the 3 cards that will pair up the board. Then you are looking for any of the 2 cards that will trip up the board. Then divide by the number of combinations. That gives you:

(48 * 3 * 2) / (50 * 49 * 48) = 0.2448979%

Holding paired cards and flopping EXACTLY four of a kind, two cards to your hole pair.

You are looking for any of the 2 cards that will trip your hole pair. Then you are looking for the remaining card that will quad your hole pair. Then you are looking for any of the remaining 48 cards. Then divide by the number of combinations. That gives you:

(2 * 1 * 48) / (50 * 49 * 48) = 0.0816327%

That is the chance of flopping the quads on the first and second flop cards. There are two other ways to flop quads that have an equal chance: (1) tripping on the first card and quading on the third, (2) tripping on the second card and quading on the third.

So, you have 3 * 0.0816327% = 0.245%