Texas Hold'em Poker Probability - Flop (5 cards)

There are 3 cards on the flop.  Including your two cards, there are total 5 cards.

Let's look at the 5 card combination first.

The total combination of 5 cards is 2,598,960= 52*51*50*49*48/5!=C(52,5)

Royal Flush: 4

Straight Flush: 36=10*4-4  from A2345 to TJQKA minus Royal Flush

Four of a Kind: 624=13*12*4   from AAAA* to KKKK*, * could be any of the other 48 cards

Full House: 3744=P(13,2)*P(4,2)  pick two different number from A to K for the 3 cards and 2 cards and with 4*3 suit combinations.

Flush: 5108=C(13,5)*4-36-4  pick five different numbers from A to K, four different suits, minus straight.

Straight: 10200=10*(4^5-4)  from A2345 to TJQKA with any kind of suit combination minus flush.

Three of a Kind: 54912=13*4*C(48,2)-3744  from AAA to KKK with 4 different suit combination, and pick any two cards from the remaining 48 cards, then minus full house.

                            54912=13*4*C(12,2)*4^2  from AAA to KKK with 4 different suit combination, and pick any two different numbers from the remaining 12 numbers (each number has 4 suit choices).

Two Pairs: 123552=C(13,2)*6*6*11*4  pick two different numbers from A to K, each number has 6 suit combination, and pick one from the remaining card without the same number(11*4).

One Pair: 1098240=13*6*C(12,3)*4^3  pick one number from A to K with 6 suit combination, then pick 3 numbers from the remaining 12 numbers (each number has 4 suit choices).

Others: 1302540=(C(13,5)-10)*(4^5-4)  pick five different numbers from A to K minus the straight combination(10), there are 4^5 suit combination minus flush(4).

The probability to get one pair or better after the flop is about 50%.

How to VERIFY: Play online poker and keep the record of the hands with the flop.  If you fold before the flop, you simply take that record out.