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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.

Texas Hold 'Em Bonus goes by the name of Easy Poker at the Casino Bregenz in Austria.

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.

The Horseshoe casino in Hammond, Indiana, follows the Las Vegas rules. Despite what the rule booklet says, the Casino Bregenz in Austria follows the Atlantic City rules.

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.

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. The lower right cell shows a house edge of 2.04%. The standard deviation is 3.51.

Texas Hold 'em Bonus Return Table — Las Vegas RulesExpand

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.

Texas Holdem Poker Hands Chart

According to Scott McIntosh the house edge is 5.5935% under the Atlantic City rules. I trust this figure 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.

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 unsuit

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