There is no doubt that Texas Hold'em Bonus Poker (THB) has a big hole-card issue. In this post I covered the situation when the AP sees one dealer hole-card, which gives the AP a 7.61% edge over the house with perfect play. Like its first cousin Ultimate Texas Hold'em, when the dealer's procedure breaks down in THB, two separate hole-cards are usually exposed. Typically, the AP will see a dealer hole-card together with either a Flop card or a River card. This post considers these two lucrative opportunities.
Running a complete cycle for each hole-carding opportunity would take approximately 310 hours to complete. Instead, I decided to approach this project by running two big Monte-Carlo simulations. In each case I assumed the AP knew his two cards and one dealer hole-card when the cards were dealt. In the first simulation, I assumed the AP also knew one of the three Flop cards. In the second simulation, I assumed the AP also knew the River card.
Known: One Dealer Hole-Card, One Flop Card
The first simulation (one dealer card, one Flop card) ran as follows. First, it picked a random starting hand for the player, a random dealer hole-card and a random Flop card. The program then cycled through the combin(48,2)*46*45*44 = 102,738,240 possible ways the hand could play out assuming the player knew those four starting cards throughout. It then computed the expected value (EV) for that starting configuration of four cards. In those situations when the EV was less than -1, the program assumed the player folded the hand and reset the EV for that hand to -1. The program played perfect hole-card strategy, but did not give any output for the strategy it determined beyond the Flop strategy. Altogether I simulated 50,000 starting hands. This computation took about 28 hours using four concurrent processes. I exported the data to an Excel spread sheet to complete the analysis and determine other statistics.
When the player knows his two cards as well as one dealer hole-card and one Flop card, then:
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The player has an edge of approximately 15.1% over the house.
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The player will fold about 21.6% of his hands.
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If the player plays all hands and never folds pre-Flop, then the player will have an edge of roughly 7.4% over the house.
Here are a few other results from the simulation:
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The hand with the most negative expected value was player = (8d, 2s), dealer hole-card = Kc, known Flop card = Kh (EV = -2.645544).
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The hand with the most positive expected value was player = (Kh, Ks), dealer hole-card = 2h, known Flop card = Kd (EV = 4.113841).
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The strongest hand that was folded was player = (Js, 6d), dealer hole-card = Ac, known Flop card = 8s (EV = -1.000024).
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The weakest hand that was played was player = (9s, 8c), dealer hole-card = Qh, known Flop card = Ac (EV = -0.999951).
Known: One Dealer Hole-Card, River Card
The second simulation (one dealer card, River card) ran as follows. First, it picked a random starting hand for the player, a random dealer hole-card and a random River card. The program then cycled through the combin(48,3)*45*44 = 34,246,080 possible ways the hand could play out assuming the player knew those four starting cards throughout. It then computed the expected value (EV) for that starting configuration of four cards. In those situations when the EV was less than -1, the program assumed the player folded the hand and reset the EV for that hand to -1. The program played perfect hole-card strategy, but did not give any output for the strategy it determined beyond the Flop strategy. Altogether I simulated 50,000 starting hands. This computation took about 18 hours using four concurrent processes. I exported the data to an Excel spread sheet to complete the analysis and determine other statistics.
When the player knows his two cards as well as one dealer hole-card and the River card, then
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The player has an edge of approximately 24.0% over the house.
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The player will fold about 16.9% of his hands.
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If the player plays all hands and never folds, then the player will have an edge of roughly 18.5% over the house.
Here are a few other results from the simulation:
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The hand with the most negative expected value was player = (9h, 2d), dealer hole-card = Kd, known River card = Kh (EV = -2.559750).
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The hand with the most positive expected value was player = (Qc, Qs), dealer hole-card = 2c, known River card = Qd (EV = 4.086408).
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The strongest hand that was folded was player = (8s, 4s), dealer hole-card = Ad, known River card = 9c (EV = -1.000037).
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The weakest hand that was played was player = (9c,5d), dealer hole-card = Qs, known River card = 3h (EV = -0.999857).
This edge when the AP knows the River card is substantially higher than the previous case. If the player knows one Flop card, then that knowledge has value until the Flop is played. In other words, it only helps the AP with his pre-Flop decision. After the Flop, knowledge of a Flop card has lost its value (the option has expired).
When the AP knows one dealer hole-card and one Flop card:
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The AP knows 0 cards when he makes his original wager.
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The AP knows 4 cards when he makes his Flop bet.
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The AP knows 6 cards when he makes his Turn bet.
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The AP knows 7 cards when he makes his River bet.
This hole-carding situation can then be described as (0,4,6,7).
By contrast, if the AP knows the River card, then this information will help him with all three wagering opportunities, effectively making the Turn and River bets strategically equivalent. The AP has two additional opportunities to exercise an option.
When the AP knows one dealer hole-card and the River card:
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The AP knows 0 cards when he makes his original wager.
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The AP knows 4 cards when he makes his Flop bet.
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The AP knows 7 cards when he makes his Turn bet.
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The AP knows 8 cards when he makes his River bet.
This hole-carding situation can then be described as (0,4,7,8). Similarly, the situation when the AP knows only the dealer's hole-card and no other cards can be described as (0,3,6,7), while the base game with no extra information is (0,2,5,6).
In summary:
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(0,2,5,6) -- House edge = 2.04%
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(0,3,6,7) -- Player edge = 7.61%
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(0,4,6,7) -- Player edge = 15.1%
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(0,4,7,8) -- Player edge = 24.0%
The game (0,4,7,7) corresponds to knowing the Turn card. This may occasionally occur, but my experience is that the card that is exposed is the bottom card of the packet. This card usually ends up being either a Flop card or the River card. There are many other tantalizing possibilities for the AP, including knowing both dealer hole-cards (0,4,7,8) and/or knowing two or more of the community cards. I am going to forgo all of this analysis.
The following Excel file contains the results from the simulations I did for this post:
THB_Hole_Card_Dealer_Community (6 M)
I have occasionally spotted dealers flashing hole-cards on this game. Like most poker-style carnival games, THB is often dealt with a carefree attitude. Though the full strategy may be too difficult to master (or even figure out), I have no doubt that common sense will lead to a huge advantage when one dealer hole-card and one community card are known.
