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

Rather than the previous 5 card combinations, players might also be interested in knowing how often we get open-ended straight or 4 card flush after the flop just shown to us.

4 Card Flush: 111540=C(13,4)*4*(52-13)  pick 4 different numbers from A to K with 4 suits, and pick one from the remaining card without the same suit(52-13).

Open Ended Straight: 60480=9*(13-6)*(4^5-4*3*5-4)  from 2345 to TJQK and pick one from the remaining number without making straight (13-6), the total suit combination 4^5 minus 4-card-flush(4*3*5) minus flush(4).

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After the flop is shown, there is about 1 in 13 players to have something better than one pair.

But if we also include the 4 card flush and open-ended straight, then it's about 1 in 7.

So if there might be 7 players into a game to see the flop, you would better be prepared to fight down through the river.

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.

Texas Hold'em Poker Probability - Starting Hand

Poker is an imperfect information game, and it's all about the theory of probability.  The key to win is based on how we can use the probability to gain an edge with some call/raise skills.

In Texas Hold'em, you get two cards first, then the flop, the turn, and the river card.  In such a process, we can find out the probability of the starting hand (2 cards), the flop(3/5 cards), the turn (4/6 cards), and the river (5/7 cards).

If we simplify it a little bit, we focus on the starting hand, the flop, and the river.  We would like to know the 2-card, 3-card, 5-card, and 7-card probability of the combination of the cards.

There are 52 cards.

For the starting hand, 

the number of the combination of the starting hand is 1,326 = 52*51/2.

pair - 78 = 52*3/2

suited - 312 = 52*12/2

others - 936 = 52*(51-3-12)/2

pair : suited : others = 1:4:12

You have about 30% probability to get a pair(6%) or suited(24%) starting hand.

How to VERIFY: You can go to play online poker and keep a record of your starting hands to verify the result.