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Last Updated: March 26, 2018

Probabilities in Two-Player Texas Hold 'Em

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


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



Written by: Michael Shackleford

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