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# 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