Texas Hold 'Em Bonus

Last update: Sep. 28, 2008

Introduction

Texas Hold 'em Bonus is a poker based table game I noticed at the Flamingo Hilton in Las Vegas in September 2005. I also hear that it can be found in Atlantic City. Like real Texas Hold 'em the player may bet his hole card, after the flop, and after the turn (but not after the river). As in one on one poker all bets pay one to one, except the ante requires a straight or flush (depending on the location) or higher to pay.

Las Vegas Rules

The game is played with a single 52-card deck.
The player makes an Ante wager, plus an optional Bonus bet.
Two hole cards are dealt face down to the player and dealer. The player may look at his own cards.
The player must either fold or make a Flop bet. The Flop bet must be two times the amount of the ante.
Three community cards (the Flop) are dealt.
The player may do nothing or make a Turn bet. The Turn bet must be exactly equal to the ante bet.
A fourth community card is dealt (the Turn).
The player may do nothing or make a River bet. The River bet must be exactly equal to the ante bet.
A fifth community card is dealt (the River).
The player and dealer each make the best five-card hand using any combination of the five community cards and his own two initial hole cards. The higher hand wins.
If the dealer has the higher hand the player will lose all wagers, except possibly the Bonus bet.
If the player has the higher hand the Flop, Turn, and River bets will pay even money. If the player has a straight or higher the Ante bet will also pay even money, otherwise it will push.
If the player and dealer have hands of equal value the the Ante, Flop, Turn, and River bets will all push.
The Bonus bet will pay based only the player's and dealer's initial cards, according to the pay table posted.

I have an unconfirmed report that the Horseshoe casino in Hammond, Indiana, follows the Las Vegas pay table.

Atlantic City Rules

There are only two rule differences under the Las Vegas rules. The player needs a flush or higher to win even money on the ante. The bonus bet does not have a line item for both player and dealer having a pair of aces.

Colorado Rules

In Colorado the law forbids betting more than $5 at a time. Texas Hold 'em Bonus is played with a $5 minimum and maximum on the ante. To be compliant with the law the flop bet is equal to the ante bet.

House Edge

The following table shows all the possible outcomes assuming optimal player strategy and the Las Vegas rules. An explanation of each column follows the table.

Texas Hold 'em Bonus Return Table - Las Vegas Rules

Turn Bet River Bet Outcome Win Combinations Probability Return

No No Dealer wins -3 16191937969988 0.291077 -0.873232

No No Tie 0 1259275163176 0.022638 0

No No Player wins with less than straight 2 5303424537200 0.095338 0.190676

No No Player wins with straight or higher 3 1084471210036 0.019495 0.058486

No Yes Dealer wins -4 1396173000700 0.025099 -0.100394

No Yes Tie 0 167704605536 0.003015 0

No Yes Player wins with less than straight 3 2296867062000 0.04129 0.12387

No Yes Player wins with straight or higher 4 605730184164 0.010889 0.043556

Yes No Dealer wins -4 1896639509356 0.034095 -0.136381

Yes No Tie 0 135581445288 0.002437 0

Yes No Player wins with less than straight 3 1247381880480 0.022424 0.067271

Yes No Player wins with straight or higher 4 252140929436 0.004533 0.018131

Yes Yes Dealer wins -5 5604708827316 0.100754 -0.50377

Yes Yes Tie 0 547943447680 0.00985 0

Yes Yes Player wins with less than straight 4 12372657154480 0.222419 0.889677

Yes Yes Player wins with straight or higher 5 2747896241164 0.049398 0.24699

Player folds -1 2517086880000 0.045249 -0.045249

Total 55627620048000 1 -0.020369

Explanation of table

Column 1:Yes if player makes turn bet, no otherwise
Column 2:Yes if player makes river bet, no otherwise
Column 3:Outcome
Column 4: Net win
Column 5:Total combinations
Column 6:Probability
Column 7:Contribution to return (win*probability)

The lower right cell shows a house edge of 2.037%. However I think this makes the game look like a worse bet than it really is. What this means is that for every dollar bet on the ante the player can expect to lose 2.037 cents. However by the end of the hand the average wager will be 3.8181 times the ante. So the element of risk, or ratio of loss to total amount bet is a much lower 2.037%/3.8181=0.5335%. The standard deviation is 3.500839.

According to Scott McIntosh the house edge is 5.5935% under the Atlantic City rules, and 2.3496% under the Colorado rules. I trust these figures because we agree exactly on the Las Vegas rules.

Strategy

The following is when the player should fold according to the rules.

Las Vegas: unsuited 2/3 to 2/7.
Atantic City: unsuited 2/3 to 2/7, and unsuited 3/4.
Colorado: never

I have not quantified the strategy past the first decision point. I can say that under the Las Vegas rules the player who made the Flop bet will make the Turn bet 43.13% of the time, and the River bet 48.46% of the time. If the player made the turn bet, then he will also make the river bet 85.76% of the time. If the player made the flop bet, but not the turn bet, he will make the river bet 15.78% of the time.

The following table shows the expected value of all 169 possible initial hands under both the Las Vegas and Atlantic City rules.

Two-Card Expected Values

Initial Hand

Las Vegas Rules

Atlantic City Rules

2,3 unsuited -1.155927 -1.195592

2,4 unsuited -1.091604 -1.13521

2,5 unsuited -1.019987 -1.067022

2,6 unsuited -1.046723 -1.077817

2,7 unsuited -1.031301 -1.050057

2,8 unsuited -0.896002 -0.917393

2,9 unsuited -0.763132 -0.784523

2,T unsuited -0.613979 -0.636507

2,J unsuited -0.459414 -0.478676

2,Q unsuited -0.28969 -0.305687

2,K unsuited -0.102668 -0.1154

2,A unsuited 0.184373 0.160096

3,4 unsuited -0.950034 -1.009069

3,5 unsuited -0.874456 -0.937452

3,6 unsuited -0.899632 -0.946711

3,7 unsuited -0.8854 -0.919748

3,8 unsuited -0.850607 -0.872617

3,9 unsuited -0.699497 -0.724139

3,T unsuited -0.549947 -0.575726

3,J unsuited -0.395206 -0.417719

3,Q unsuited -0.225563 -0.24481

3,K unsuited -0.038499 -0.054482

3,A unsuited 0.254437 0.226376

4,5 unsuited -0.735126 -0.813569

4,6 unsuited -0.756096 -0.819136

4,7 unsuited -0.740561 -0.790904

4,8 unsuited -0.7068 -0.744413

4,9 unsuited -0.654394 -0.679669

4,T unsuited -0.487036 -0.516079

4,J unsuited -0.332233 -0.358011

4,Q unsuited -0.162612 -0.185124

4,K unsuited 0.024799 0.005551

4,A unsuited 0.321642 0.290174

5,6 unsuited -0.614318 -0.692805

5,7 unsuited -0.595481 -0.661787

5,8 unsuited -0.560487 -0.614096

5,9 unsuited -0.508593 -0.549471

5,T unsuited -0.437507 -0.467184

5,J unsuited -0.265364 -0.294408

5,Q unsuited -0.095642 -0.12142

5,K unsuited 0.092367 0.069855

5,A unsuited 0.392893 0.358138

6,7 unsuited -0.457726 -0.536213

6,8 unsuited -0.421014 -0.48732

6,9 unsuited -0.36839 -0.421999

6,T unsuited -0.297258 -0.339273

6,J unsuited -0.217975 -0.244386

6,Q unsuited -0.031216 -0.056994

6,K unsuited 0.156819 0.134306

6,A unsuited 0.370719 0.351472

7,8 unsuited -0.284039 -0.362526

7,9 unsuited -0.229986 -0.296292

7,T unsuited -0.157768 -0.212535

7,J unsuited -0.078216 -0.116966

7,Q unsuited 0.019887 -0.003259

7,K unsuited 0.224911 0.202399

7,A unsuited 0.456798 0.434885

8,9 unsuited -0.092377 -0.170864

8,T unsuited -0.016383 -0.08401

8,J unsuited 0.062709 0.011207

8,Q unsuited 0.16144 0.125955

8,K unsuited 0.282809 0.262927

8,A unsuited 0.533716 0.511803

9,T unsuited 0.127027 0.046521

9,J unsuited 0.20434 0.139978

9,Q unsuited 0.304465 0.256228

9,K unsuited 0.428187 0.395968

9,A unsuited 0.602407 0.583126

T,J unsuited 0.370279 0.288219

T,Q unsuited 0.471694 0.405776

T,K unsuited 0.600228 0.550453

T,A unsuited 0.78194 0.745643

J,Q unsuited 0.535555 0.472903

J,K unsuited 0.665761 0.619251

J,A unsuited 0.852493 0.819462

Q,K unsuited 0.735444 0.6922

Q,A unsuited 0.925311 0.895545

K,A unsuited 0.999317 0.972816

2,3 suited -0.886985 -0.923987

2,4 suited -0.826008 -0.866658

2,5 suited -0.75798 -0.801803

2,6 suited -0.782881 -0.811796

2,7 suited -0.768076 -0.785429

2,8 suited -0.639021 -0.658814

2,9 suited -0.51015 -0.529944

2,T suited -0.364595 -0.385438

2,J suited -0.212709 -0.23053

2,Q suited -0.044946 -0.059747

2,K suited 0.141142 0.129363

2,A suited 0.425022 0.402419

3,4 suited -0.693044 -0.74813

3,5 suited -0.62123 -0.679983

3,6 suited -0.644776 -0.688639

3,7 suited -0.631082 -0.663007

3,8 suited -0.597478 -0.617842

3,9 suited -0.450882 -0.473684

3,T suited -0.304918 -0.328768

3,J suited -0.152868 -0.173698

3,Q suited 0.014889 -0.002919

3,K suited 0.201376 0.186589

3,A suited 0.490618 0.464512

4,5 suited -0.490199 -0.563403

4,6 suited -0.509707 -0.5685

4,7 suited -0.494853 -0.541737

4,8 suited -0.461634 -0.496581

4,9 suited -0.40913 -0.432516

4,T suited -0.246108 -0.272979

4,J suited -0.094025 -0.117875

4,Q suited 0.073849 0.053019

4,K suited 0.260929 0.243121

4,A suited 0.553554 0.524295

5,6 suited -0.376191 -0.449435

5,7 suited -0.357989 -0.419804

5,8 suited -0.323018 -0.372923

5,9 suited -0.270833 -0.308801

5,T suited -0.199974 -0.227429

5,J suited -0.031373 -0.058245

5,Q suited 0.136745 0.112895

5,K suited 0.324793 0.303963

5,A suited 0.620767 0.588467

6,7 suited -0.227072 -0.300316

6,8 suited -0.190276 -0.252091

6,9 suited -0.137464 -0.187369

6,T suited -0.066542 -0.105559

6,J suited 0.013273 -0.011162

6,Q suited 0.197628 0.173777

6,K suited 0.385858 0.365029

6,A suited 0.59982 0.582012

7,8 suited -0.05986 -0.133104

7,9 suited -0.005551 -0.067365

7,T suited 0.066287 0.015313

7,J suited 0.146267 0.110272

7,Q suited 0.246024 0.22461

7,K suited 0.451106 0.430277

7,A suited 0.682911 0.662637

8,9 suited 0.125648 0.052404

8,T suited 0.200955 0.13792

8,J suited 0.280795 0.232842

8,Q suited 0.381333 0.348359

8,K suited 0.50656 0.488168

8,A suited 0.757219 0.736945

9,T suited 0.33757 0.262459

9,J suited 0.416267 0.356253

9,Q suited 0.518407 0.473476

9,K suited 0.646145 0.616192

9,A suited 0.822276 0.80444

T,J suited 0.574533 0.497955

T,Q suited 0.677017 0.615537

T,K suited 0.808521 0.762138

T,A suited 0.990289 0.956538

J,Q suited 0.737666 0.679207

J,K suited 0.870492 0.82713

J,A suited 1.056211 1.02548

Q,K suited 0.936506 0.896165

Q,A suited 1.124432 1.096723

K,A suited 1.194024 1.169336

2,2 pair -0.134137 -0.140431

3,3 pair 0.078584 0.068777

4,4 pair 0.322289 0.308953

5,5 pair 0.589457 0.572593

6,6 pair 0.837465 0.820601

7,7 pair 1.093669 1.076805

8,8 pair 1.353048 1.336184

9,9 pair 1.612114 1.59525

T,T pair 1.87899 1.860933

J,J pair 2.097415 2.082887

Q,Q pair 2.315838 2.304838

K,K pair 2.535691 2.528219

A,A pair 2.788899 2.78206

Bonus Bet

The bonus bet pays based only on the players hole cards, and in some cases the dealer's two hole cards. Who wins the hand does not matter. There seem to be two pay tables available for the bonus bet as follows. From what I hear the Atlantic City pay table is also used in Missouri and Indiana.

Bonus Bet - Las Vegas Rules

Event

Pays

Combinations

Probability

Return

Player has AA, dealer has AA 1000 6 0.000004 0.003694

Player has AA 30 7344 0.004521 0.135636

Player has AK suited 25 4900 0.003017 0.075415

Player has AQ or AJ suited 20 9800 0.006033 0.120664

Player has AK unsuited 15 14700 0.00905 0.135747

Player has KK, QQ, or JJ 10 22050 0.013575 0.135747

Player has AQ or AJ unsuited 5 29400 0.0181 0.090498

Player has Pair 22 to 10,10 3 66150 0.040724 0.122172

All other -1 1470000 0.904977 -0.904977

Total 1624350 1 -0.085406

Bonus Bet - Atlantic City Rules

Event

Pays

Combinations

Probability

Return

AA 30 7350 0.004525 0.135747

AK suited 25 4900 0.003017 0.075415

AQ or AJ suited 20 9800 0.006033 0.120664

AK unsuited 15 14700 0.00905 0.135747

KK, QQ, or JJ 10 22050 0.013575 0.135747

AQ or AJ unsuited 5 29400 0.0181 0.090498

Pair 22 to 10,10 3 66150 0.040724 0.122172

All other -1 1470000 0.904977 -0.904977

Total 1624350 1 -0.088989

The lower right cells show a house edge on the bonus bet of 8.5406% under the Las Vegas rules and 8.8989% under the Atlantic City rules.

Methodology

A brute force approach was used to cycle through all possible cominations of cards. In total 7,089,794,712,000 hands were played out, requiring 72 days of computer time to loop through them all.

Acknowledgements

I would like to thank Scott McIntosh of Online Poker Room Reviews for his contributions to this page. He confirmed my Las Vegas results and provided the Atlantic City house edge and expected values. His program runs in six hours compared to the 72 days mine takes, so I was happy to use his numbers. For reviews and advice on playing poker online I encourage you to visit his web site.

Links

There is a good Texas Hold 'Em Bonus calculator at beatingbonuses.com.

Analysis at ReviewPokerRooms.com.

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