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Probabilities in TwoPlayer 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 twoplayer 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