Texas Hold’em Poker is one of the most popular card games, especially among betting games. While poker is played in a multitude of variations, Texas Hold’em is the version played most often at casinos and is the most popular among the “community cards” variants of poker. It is also the variant played at the World Series of Poker. On the poker hand rankings chart, a Straight weighs in at fifth position. Directly above a straight is a Flush poker hand. The best flush you can form is an Ace-high Flush. In a standard game of Texas Hold’em poker, a Straight is still a great hand to form. There are several hands that rank beneath a Straight, notably Three of a Kind. Texas Hold'em Poker Runner-Runner (Backdoor) Odds. When it takes two perfect cards on both the turn and the river to covert an average or weak hand after the flop to a strong one, this should normally qualify as a 'long shot'. Also, 'strong' hands are considered here to be straights and better.

“I couldn’t fold, I had an up & down straight draw on the flop!” An up and down straight draw, or an open ended straight draw means that you have eight cards that can complete your straight. You’ll complete your straight 31.5% of the time on the river – and what a sweet feeling that is! Legalizing Texas holdem and other forms of poker in many places, including online. The answer to this question boils down to the mathematics behind the game. If the math shows one player can win more often than another based on the mathematical and statistical truths about Texas.

Introduction

This page examines the probabilities of each final hand of an arbitrary player, referred to as player two, given the poker value of the hand of the other player, referred to as player one. Combinations shown are out of a possible combin(52,5)×combin(47,2)×combin(45,2) = 2,781,381,002,400. The primary reason for this page was to assist with bad beat probabilities in a two-player game, for example the Bad Beat Bonus in Ultimate Texas Hold 'Em.

For example, if you wish to know the probability of a particular player getting a full house and losing to a four of a kind, we can see from table 7 that there are 966,835,584 such combinations. The same table shows us that given that player one has a full house, the probability of losing to a four of a kind is 0.013390. To get the probability before any cards are dealt, divide 966,835,584 by the total possible combinations of 2,781,381,002,400, which yields 0.0002403.

Table 1 shows the number of combinations for each hand of a second player, given that the first player has less than a pair.

Table 1 — First Player has Less than Pair

Event	Pays	Probability
Less than pair	164,934,908,760	0.340569
Pair	228,994,769,160	0.472845
Two pair	43,652,558,880	0.090137
Three of a kind	7,303,757,580	0.015081
Straight	26,248,866,180	0.054201
Flush	13,060,678,788	0.026969
Full house	-	0.000000
Four of a kind	-	0.000000
Straight flush	85,751,460	0.000177
Royal flush	10,532,592	0.000022
Total	484,291,823,400	1.000000

Table 2 shows the number of combinations for each hand of a second player, given that the first player has a pair.

Table 2 — First Player has a Pair

Event	Pays	Probability
Less than pair	228,994,769,160	0.187874
Pair	574,484,133,960	0.471324
Two pair	270,127,833,552	0.221621
Three of a kind	47,736,401,832	0.039164
Straight	50,797,137,096	0.041676
Flush	30,076,271,352	0.024675
Full house	15,829,506,000	0.012987
Four of a kind	586,278,000	0.000481
Straight flush	214,250,184	0.000176
Royal flush	25,380,864	0.000021
Total	1,218,871,962,000	1.000000

Table 3 shows the number of combinations for each hand of a second player, given that the first player has a two pair.

Table 3 — First Player has a Two Pair

Event	Pays	Probability
Less than pair	43,652,558,880	0.066798
Pair	270,127,833,552	0.413355
Two pair	246,286,292,328	0.376872
Three of a kind	31,155,189,408	0.047674
Straight	18,549,991,152	0.028386
Flush	14,200,694,712	0.021730
Full house	28,751,944,680	0.043997
Four of a kind	653,378,400	0.001000
Straight flush	109,829,304	0.000168
Royal flush	12,673,584	0.000019
Total	653,500,386,000	1.000000

Table 4 shows the number of combinations for each hand of a second player, given that the first player has a three of a kind.

Table 4 — First Player has a Three of a Kind

Event	Pays	Probability
Less than pair	7,303,757,580	0.054369
Pair	47,736,401,832	0.355348
Two pair	31,155,189,408	0.231918
Three of a kind	27,586,332,384	0.205352
Straight	3,310,535,196	0.024643
Flush	2,606,403,900	0.019402
Full house	12,910,316,760	0.096104
Four of a kind	1,705,867,680	0.012698
Straight flush	19,970,844	0.000149
Royal flush	2,304,216	0.000017
Total	134,337,079,800	1.000000

Table 5 shows the number of combinations for each hand of a second player, given that the first player has a straight.

Table 5 — First Player has a Straight

Event	Pays	Probability
Less than pair	26,248,866,180	0.204299
Pair	50,797,137,096	0.395362
Two pair	18,549,991,152	0.144377
Three of a kind	3,310,535,196	0.025766
Straight	25,219,094,136	0.196284
Flush	3,229,836,828	0.025138
Full house	975,510,000	0.007593
Four of a kind	43,198,800	0.000336
Straight flush	98,961,348	0.000770
Royal flush	9,485,064	0.000074
Total	128,482,615,800	1.000000

Table 6 shows the number of combinations for each hand of a second player, given that the first player has a flush.

Table 6 — First Player has a Flush

Event	Pays	Probability
Less than pair	13,060,678,788	0.155206
Pair	30,076,271,352	0.357410
Two pair	14,200,694,712	0.168754
Three of a kind	2,606,403,900	0.030973
Straight	3,229,836,828	0.038382
Flush	19,608,838,592	0.233021
Full house	1,102,206,960	0.013098
Four of a kind	50,221,200	0.000597
Straight flush	191,762,164	0.002279
Royal flush	23,604,264	0.000281
Total	84,150,518,760	1.000000

Table 7 shows the number of combinations for each hand of a second player, given that the first player has a full house.

Table 7 — First Player has a Full House

Event	Pays	Probability
Less than pair	-	0.000000
Pair	15,829,506,000	0.219222
Two pair	28,751,944,680	0.398185
Three of a kind	12,910,316,760	0.178795
Straight	975,510,000	0.013510
Flush	1,102,206,960	0.015264
Full house	11,661,414,336	0.161499
Four of a kind	966,835,584	0.013390
Straight flush	8,767,440	0.000121
Royal flush	993,600	0.000014
Total	72,207,495,360	1.000000

Table 8 shows the number of combinations for each hand of a second player, given that the first player has a four of a kind.

Table 8 — First Player has a Four of a Kind

Event	Pays	Probability
Less than pair	-	0.000000
Pair	586,278,000	0.125418
Two pair	653,378,400	0.139772
Three of a kind	1,705,867,680	0.364923
Straight	43,198,800	0.009241
Flush	50,221,200	0.010743
Full house	966,835,584	0.206828
Four of a kind	668,375,136	0.142980
Straight flush	390,960	0.000084
Royal flush	44,160	0.000009
Total	4,674,589,920	1.000000

Table 9 shows the number of combinations for each hand of a second player, given that the first player has a straight flush.

Table 9 — First Player has a Straight Flush

Event	Pays	Probability
Less than pair	85,751,460	0.110699
Pair	214,250,184	0.276582
Two pair	109,829,304	0.141782
Three of a kind	19,970,844	0.025781
Straight	98,961,348	0.127752
Flush	191,762,164	0.247552
Full house	8,767,440	0.011318
Four of a kind	390,960	0.000505
Straight flush	44,354,840	0.057259
Royal flush	596,856	0.000770
Total	774,635,400	1.000000

Table 10 shows the number of combinations for each hand of a second player, given that the first player has a royal flush.

Table 10 — First Player has a Royal Flush

Event	Pays	Probability
Less than pair	10,532,592	0.117164
Pair	25,380,864	0.282336
Two pair	12,673,584	0.140981
Three of a kind	2,304,216	0.025632
Straight	9,485,064	0.105512
Flush	23,604,264	0.262573
Full house	993,600	0.011053
Four of a kind	44,160	0.000491
Straight flush	596,856	0.006639
Royal flush	4,280,760	0.047619
Total	89,895,960	1.000000

The following table shows the number of combinations for each hand of player 1 by the winner of the hand.

Table 11 — Winning Player by Hand of Player 1 — Combinations

Player 1	Win	Tie	Loss
Less than pair	76,626,795,600	11,681,317,560	395,983,710,240	484,291,823,400
Pair	496,857,988,764	38,757,694,752	683,256,278,484	1,218,871,962,000
Two pair	419,896,266,012	34,054,545,168	199,549,574,820	653,500,386,000
Three of a kind	97,664,829,948	4,647,370,128	32,024,879,724	134,337,079,800
Straight	103,685,076,072	15,662,001,240	9,135,538,488	128,482,615,800
Flush	71,523,195,288	2,910,219,176	9,717,104,296	84,150,518,760
Full house	62,810,500,464	5,179,382,208	4,217,612,688	72,207,495,360
Four of a kind	4,240,864,800	198,204,864	235,520,256	4,674,589,920
Straight flush	734,237,144	35,247,960	5,150,296	774,635,400
Royal flush	85,615,200	4,280,760	-	89,895,960
Total	1,334,125,369,292	113,130,263,816	1,334,125,369,292	2,781,381,002,400

Texas Holdem Probability Equation

The following table shows the probability for each hand of player 1 by the winner of the hand. The bottom row shows that each player has a 47.97% chance of winning and a 4.07% chance of a tie.

Table 12 — Winning Player by Hand of Player 1 — Probabilities

Player 1 Hand	Player 1	Tie	Player 2	Total
Less than pair	0.027550	0.004200	0.142369	0.174119
Pair	0.178637	0.013935	0.245654	0.438225
Two pair	0.150967	0.012244	0.071745	0.234955
Three of a kind	0.035114	0.001671	0.011514	0.048299
Straight	0.037278	0.005631	0.003285	0.046194
Flush	0.025715	0.001046	0.003494	0.030255
Full house	0.022582	0.001862	0.001516	0.025961
Four of a kind	0.001525	0.000071	0.000085	0.001681
Straight flush	0.000264	0.000013	0.000002	0.000279
Royal flush	0.000031	0.000002	0.000000	0.000032
Total	0.479663	0.040674	0.479663	1.000000

Texas Holdem Probability Calculator

Written by: Michael Shackleford