Probabilities in Two-Player Texas Hold 'Em
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
|Less than pair||164,934,908,760||0.340569|
|Three of a kind||7,303,757,580||0.015081|
|Four of a kind||-||0.000000|
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
|Less than pair||228,994,769,160||0.187874|
|Three of a kind||47,736,401,832||0.039164|
|Four of a kind||586,278,000||0.000481|
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
|Less than pair||43,652,558,880||0.066798|
|Three of a kind||31,155,189,408||0.047674|
|Four of a kind||653,378,400||0.001000|
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
|Less than pair||7,303,757,580||0.054369|
|Three of a kind||27,586,332,384||0.205352|
|Four of a kind||1,705,867,680||0.012698|
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
|Less than pair||26,248,866,180||0.204299|
|Three of a kind||3,310,535,196||0.025766|
|Four of a kind||43,198,800||0.000336|
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
|Less than pair||13,060,678,788||0.155206|
|Three of a kind||2,606,403,900||0.030973|
|Four of a kind||50,221,200||0.000597|
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
|Less than pair||-||0.000000|
|Three of a kind||12,910,316,760||0.178795|
|Four of a kind||966,835,584||0.013390|
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
|Less than pair||-||0.000000|
|Three of a kind||1,705,867,680||0.364923|
|Four of a kind||668,375,136||0.142980|
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
|Less than pair||85,751,460||0.110699|
|Three of a kind||19,970,844||0.025781|
|Four of a kind||390,960||0.000505|
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
|Less than pair||10,532,592||0.117164|
|Three of a kind||2,304,216||0.025632|
|Four of a kind||44,160||0.000491|
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
|Less than pair||76,626,795,600||11,681,317,560||395,983,710,240||484,291,823,400|
|Three of a kind||97,664,829,948||4,647,370,128||32,024,879,724||134,337,079,800|
|Four of a kind||4,240,864,800||198,204,864||235,520,256||4,674,589,920|
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|
|Three of a kind||0.035114||0.001671||0.011514||0.048299|
|Four of a kind||0.001525||0.000071||0.000085||0.001681|
Written by: Michael Shackleford