Texas Holdem Rechner
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Introduction
Heads Up Hold 'Em is an Ultimate Texas Hold 'Em variant by Galaxy Gaming, based on Texas Hold 'Em. The player may raise his bet one time, and has three opportunities to do so. The earlier he raises, the more he can bet. The main differences between Heads Up Hold 'Em and Ultimate Texas Hold 'Em are in the former the player may raise only 3x his Ante bet before the flop, but the game includes bad beat bonuses for losing with a straight or higher.
- Welcome to Simple Poker! Here you can find advanced GTO poker solvers for calculating game theory optimal strategies. On our website, besides poker GTO solvers you can find packages with already calculated optimal strategies for various poker games, general information about GTO, video tutorials, tools for learning GTO and documentation.
- You must understand what all the poker hands are when playing Texas Hold'em. To get you up to speed, here's a quick rundown of hand order. Royal Flush - 10, J, K, Q, A of the same suit.
Rules
Following are the rules for Heads Up Hold 'Em. For those used to the terminology in Ultimate Texas Hold 'Em, what is called the Blind there called the Odds bet here.
- The game is played with a single ordinary 52-card deck.
- The player must make an equal bet on both the Ante and Odds.
- Two cards are dealt face down to the player and dealer. The player may look at his own cards.
- The player can check or make a Play bet equal to three times the Ante.
- The dealer turns over three community cards.
- If the player previously checked, then he may make a Play bet equal to two times his Ante or check again. If the player already made a Play bet, then he may not bet further.
- Two final community cards are turned over.
- If the player previously checked twice, then he must either make a Play bet equal to exactly his Ante, or fold, losing both his Ante and Odds bets. If the player already raised he may not bet further.
- The player and dealer will both make the best possible hand using any combination of their own two cards and the five community cards.
- The dealer will need at least a pair to open.
- The following table shows how the Ante, Odds, and Play bets are scored, according to who wins, and whether the dealer opens.
Scoring Rules
Winner Dealer Opens Ante Play Odds Player Yes Win Win See rule 12 Player No Push Win See rule 12 Dealer Yes Lose Lose See rule 12 Dealer No Push Lose Lose Tie Yes or No Push Push Push - Winning Ante and Play bets pay 1 to 1. Winning Odds bets pay according to value of the hand and whether it wins or loses. The following pay table shows what winning Odds bets pay.
Winning Odds Bet Pay Table
Hand Pays Royal Flush 500 Straight Flush 50 Quads 10 Full House 3 Flush 1.5 Straight 1 All Other Push Galaxy Gaming, the game owner, has four different pay tables for losing blind bets, as follows.
Losing Odds Bet Pay Table
Hand Pay Table 1 2 3 4 Straight Flush 500 500 500 500 Quads 50 50 50 25 Full House 10 10 10 6 Flush 8 6 5 5 Straight 5 5 4 4 All Other Loss Loss Loss Loss - In addition, there are two side bets, that pay based on the player's cards only, the Trips Plus and Pocket Bonus. These side bets are explained after the analysis of the base game.
Strategy
The player should make the 3X raise with any pair except deuces. Otherwise, use the following table shows when to make the large 3X raise.
The strategy for the medium and small raise are the same as in Ultimate Texas Hold 'Em, as follows:
Medium Raise: Make the 2X raise with any of the following:
- Two pair or better.
- Hidden pair*, except pocket deuces.
- Four to a flush including a hidden 10 or better to that flush
* Hidden pair = Any pair with at least one card in your hole cards (thus the pair is hidden to the dealer).
Small Raise: Make the 1X raise with any of the following, otherwise fold:
- Hidden pair or better.
- Less than 21 dealer outs beat you.
For a more powerful small and medium raise strategy, I recommend the James Grossjean strategy card for Ultimate Texas Hold 'Em.
Analysis
The following table shows all possible outcomes of each hand, what it pays, the probability, and contribution to the return under the losing Odds bet pay table number 1 (the one that goes 500-50-10-8-5). The lower right cell shows a house edge of 2.36%.
Return TableExpand
| Player | Raise | Dealer Qualifies | Winner | Pays | Combinations | Probability | Return |
|---|---|---|---|---|---|---|---|
| Fold | -2 | 5,498,078,560,920 | 0.197674 | -0.395349 | |||
| Less than pair | 1 | No | Dealer | -2 | 60,518,663,424 | 0.002176 | -0.004352 |
| Straight flush | 1 | Yes | Dealer | 498 | 20,279,100 | 0.000001 | 0.000363 |
| Four of a kind | 1 | Yes | Dealer | 48 | 1,726,735,980 | 0.000062 | 0.002980 |
| Full house | 1 | Yes | Dealer | 8 | 10,082,720,220 | 0.000363 | 0.002900 |
| Flush | 1 | Yes | Dealer | 6 | 49,072,032,216 | 0.001764 | 0.010586 |
| Straight | 1 | Yes | Dealer | 3 | 47,952,010,720 | 0.001724 | 0.005172 |
| Less than straight | 1 | Yes | Dealer | -3 | 3,006,630,550,164 | 0.108098 | -0.324295 |
| Anything | 1 | Y/N | Push | 0 | 455,081,939,824 | 0.016362 | 0.000000 |
| Royal flush | 1 | No | Player | 501 | 6,914,880 | 0.000000 | 0.000125 |
| Straight flush | 1 | No | Player | 51 | 279,004,320 | 0.000010 | 0.000512 |
| Four of a kind | 1 | No | Player | 11 | - | 0.000000 | 0.000000 |
| Full house | 1 | No | Player | 4 | - | 0.000000 | 0.000000 |
| Flush | 1 | No | Player | 2.5 | 43,096,215,600 | 0.001549 | 0.003874 |
| Straight | 1 | No | Player | 2 | 145,034,240,580 | 0.005214 | 0.010429 |
| Less than straight | 1 | No | Player | 1 | 816,981,676,824 | 0.029373 | 0.029373 |
| Royal flush | 1 | Yes | Player | 502 | 46,580,760 | 0.000002 | 0.000841 |
| Straight flush | 1 | Yes | Player | 52 | 2,023,968,588 | 0.000073 | 0.003784 |
| Four of a kind | 1 | Yes | Player | 12 | 964,337,328 | 0.000035 | 0.000416 |
| Full house | 1 | Yes | Player | 5 | 46,108,374,192 | 0.001658 | 0.008289 |
| Flush | 1 | Yes | Player | 3.5 | 202,016,746,236 | 0.007263 | 0.025421 |
| Straight | 1 | Yes | Player | 3 | 421,268,280,080 | 0.015146 | 0.045438 |
| Less than straight | 1 | Yes | Player | 2 | 1,600,861,520,204 | 0.057556 | 0.115113 |
| Less than pair | 2 | No | Dealer | -3 | 11,023,268,784 | 0.000396 | -0.001189 |
| Straight flush | 2 | Yes | Dealer | 497 | 16,724,460 | 0.000001 | 0.000299 |
| Four of a kind | 2 | Yes | Dealer | 47 | 311,575,460 | 0.000011 | 0.000527 |
| Full house | 2 | Yes | Dealer | 7 | 18,705,932,580 | 0.000673 | 0.004708 |
| Flush | 2 | Yes | Dealer | 5 | 27,615,003,664 | 0.000993 | 0.004964 |
| Straight | 2 | Yes | Dealer | 2 | 21,031,977,440 | 0.000756 | 0.001512 |
| Less than straight | 2 | Yes | Dealer | -4 | 1,979,644,169,384 | 0.071175 | -0.284699 |
| Anything | 2 | Y/N | Push | 0 | 214,144,135,720 | 0.007699 | 0.000000 |
| Royal flush | 2 | No | Player | 502 | 11,938,680 | 0.000000 | 0.000215 |
| Straight flush | 2 | No | Player | 52 | 397,598,400 | 0.000014 | 0.000743 |
| Four of a kind | 2 | No | Player | 12 | - | 0.000000 | 0.000000 |
| Full house | 2 | No | Player | 5 | - | 0.000000 | 0.000000 |
| Flush | 2 | No | Player | 3.5 | 45,718,738,920 | 0.001644 | 0.005753 |
| Straight | 2 | No | Player | 3 | 53,068,201,380 | 0.001908 | 0.005724 |
| Less than straight | 2 | No | Player | 2 | 1,207,385,216,712 | 0.043410 | 0.086819 |
| Royal flush | 2 | Yes | Player | 503 | 147,692,880 | 0.000005 | 0.002671 |
| Straight flush | 2 | Yes | Player | 53 | 3,016,851,612 | 0.000108 | 0.005749 |
| Four of a kind | 2 | Yes | Player | 13 | 20,440,911,312 | 0.000735 | 0.009554 |
| Full house | 2 | Yes | Player | 6 | 320,575,227,408 | 0.011526 | 0.069155 |
| Flush | 2 | Yes | Player | 4.5 | 183,447,763,404 | 0.006596 | 0.029680 |
| Straight | 2 | Yes | Player | 4 | 158,035,798,360 | 0.005682 | 0.022728 |
| Less than straight | 2 | Yes | Dealer | 3 | 2,415,318,761,280 | 0.086839 | 0.260516 |
| Less than pair | 3 | No | Dealer | -4 | 66,873,993,600 | 0.002404 | -0.009617 |
| Straight flush | 3 | Yes | Dealer | 496 | 14,499,400 | 0.000001 | 0.000259 |
| Four of a kind | 3 | Yes | Dealer | 46 | 316,891,120 | 0.000011 | 0.000524 |
| Full house | 3 | Yes | Dealer | 6 | 13,387,474,080 | 0.000481 | 0.002888 |
| Flush | 3 | Yes | Dealer | 4 | 20,484,007,080 | 0.000736 | 0.002946 |
| Straight | 3 | Yes | Dealer | 1 | 22,371,396,720 | 0.000804 | 0.000804 |
| Less than straight | 3 | Yes | Dealer | -5 | 3,136,124,565,400 | 0.112754 | -0.563771 |
| Anything | 3 | Y/N | Push | 0 | 223,641,379,520 | 0.008041 | 0.000000 |
| Royal flush | 3 | No | Player | 503 | 86,472,360 | 0.000003 | 0.001564 |
| Straight flush | 3 | No | Player | 53 | 180,911,880 | 0.000007 | 0.000345 |
| Four of a kind | 3 | No | Player | 13 | - | 0.000000 | 0.000000 |
| Full house | 3 | No | Player | 6 | - | 0.000000 | 0.000000 |
| Flush | 3 | No | Player | 4.5 | 41,791,833,360 | 0.001503 | 0.006762 |
| Straight | 3 | No | Player | 4 | 64,386,219,840 | 0.002315 | 0.009260 |
| Less than straight | 3 | No | Player | 3 | 1,317,173,128,560 | 0.047357 | 0.142070 |
| Royal flush | 3 | Yes | Player | 504 | 556,552,440 | 0.000020 | 0.010085 |
| Straight flush | 3 | Yes | Player | 54 | 1,444,036,640 | 0.000052 | 0.002804 |
| Four of a kind | 3 | Yes | Player | 14 | 21,003,399,360 | 0.000755 | 0.010572 |
| Full house | 3 | Yes | Player | 7 | 261,421,403,040 | 0.009399 | 0.065793 |
| Flush | 3 | Yes | Player | 5.5 | 199,160,655,360 | 0.007160 | 0.039383 |
| Straight | 3 | Yes | Player | 5 | 195,058,020,480 | 0.007013 | 0.035065 |
| Less than straight | 3 | Yes | Player | 4 | 3,140,424,343,760 | 0.112909 | 0.451635 |
| Total | 27,813,810,024,000 | 1.000000 | -0.023584 | ||||
As shown in the lower right cell, the house edge is 2.36%. This is the expected loss to the Ante wager only. For example, if the player bets $5 on both the Ante and Odds, then his expected loss would be $5 × 0.023584 = 11.79¢.
The average final wager per hand is 3.67 units. That makes the element of risk 2.36%/3.67 = 0.64%. This means for every dollar you wager in the game, on anything, other than the side bets, you can expect to lose 0.64¢.
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By comparison, the element of risk in Ultimate Texas Hold 'Em is 0.53%.
The standard deviation, relative to to the Ante bet, is 4.56.
The next table shows the house edge and element of risk under all four losing Odds bet pay tables according to the pay table for a losing Odds bet.
House Edge Summary
| Hand | Losing Odds Bet Pay Table | |||
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | |
| Straight Flush | 500 | 500 | 500 | 500 |
| Quads | 50 | 50 | 50 | 25 |
| Full House | 10 | 10 | 10 | 6 |
| Flush | 8 | 6 | 5 | 5 |
| Straight | 5 | 5 | 4 | 4 |
| All Other | Loss | Loss | Loss | Loss |
| House edge | 2.36% | 3.06% | 3.73% | 4.55% |
| Element of Risk | 0.64% | 0.83% | 1.02% | 1.24% |
This is full table cloth.
Trips Plus
The Trips Plus bet will pay according to the poker value of the player's hand regardless of the value of the dealer's hand. Following is an analysis of the most common pay table.
Trips Plus Return Table
| Hand | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Royal flush | 100 | 4,324 | 0.000032 | 0.003232 |
| Straight flush | 40 | 37,260 | 0.000279 | 0.011140 |
| Four of a kind | 30 | 224,848 | 0.001681 | 0.050420 |
| Full house | 8 | 3,473,184 | 0.025961 | 0.207688 |
| Flush | 7 | 4,047,644 | 0.030255 | 0.211785 |
| Straight | 4 | 6,180,020 | 0.046194 | 0.184775 |
| Three of a kind | 3 | 6,461,620 | 0.048299 | 0.144896 |
| All other | -1 | 113,355,660 | 0.847300 | -0.847300 |
| Total | 133,784,560 | 1.000000 | -0.033363 |
The next table shows four known pay tables for the Trips Plus.
Trips Plus Pay Tables
| Hand | Pay Table | |||
|---|---|---|---|---|
| 1 | 2 | 3 | 4 | |
| Royal flush | 100 | 100 | 100 | 100 |
| Straight flush | 40 | 40 | 40 | 40 |
| Four of a kind | 30 | 30 | 30 | 30 |
| Full house | 9 | 8 | 8 | 7 |
| Flush | 7 | 6 | 7 | 6 |
| Straight | 4 | 5 | 4 | 5 |
| Three of a kind | 3 | 3 | 3 | 3 |
| All other | -1 | -1 | -1 | -1 |
| Total | -0.74% | -1.74% | -3.34% | -4.34% |
Pocket Bonus
The Pocket Bonus bet will pay according to the value of the player's two hole cards. The follow tables show what each two cards pays, the probability, and contribution to the total return for each known pay table for the Pocket Bonus.
Pocket Bonus — Pay Table 1
| Hand | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Pair of aces | 30 | 6 | 0.004525 | 0.135747 |
| Ace & face suited | 20 | 12 | 0.009050 | 0.180995 |
| Ace & face unsuited | 10 | 36 | 0.027149 | 0.271493 |
| Pair 2s - Ks | 5 | 72 | 0.054299 | 0.271493 |
| Loser | -1 | 1,200 | 0.904977 | -0.904977 |
| Total | 1,326 | 1.000000 | -0.045249 |
Pocket Bonus — Pay Table 2
| Hand | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Pair of aces | 25 | 6 | 0.004525 | 0.113122 |
| Ace & face suited | 20 | 12 | 0.009050 | 0.180995 |
| Ace & face unsuited | 10 | 36 | 0.027149 | 0.271493 |
| Pair 2s - Ks | 5 | 72 | 0.054299 | 0.271493 |
| Loser | -1 | 1,200 | 0.904977 | -0.904977 |
| Total | 1,326 | 1.000000 | -0.067873 |
Pocket Bonus — Pay Table 3
| Hand | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Pair of aces | 30 | 6 | 0.004525 | 0.135747 |
| Ace & face suited | 20 | 12 | 0.009050 | 0.180995 |
| Ace & face unsuited | 10 | 36 | 0.027149 | 0.271493 |
| Pair 2s - Ks | 4 | 72 | 0.054299 | 0.217195 |
| Loser | -1 | 1,200 | 0.904977 | -0.904977 |
| Total | 1,326 | 1.000000 | -0.099548 |
Acknowledgments
I would like to thank Charles Mousseau and Stephen How for their assistance, which confirmed my analysis. Thanks to Charles also for his advice on the medium and small raise strategy.
Internal Links
Detailed calculations on some starting hands.
Extneral Links
Play Heads Up Hold 'Em at the Galaxy Gaming web site. One click and you're playing.