Casino Hold 'em Poker

Introduction

Casino Hold 'Em is a poker variation, similar to Texas Hold 'Em, that is played in many countries of the world.

The game also goes by the name Casino Hold 'Em Open. Under this name, the rules are the same except the player hole cards are dealt face up.
In normal Casino Hold 'Em they are dealt face down.

Rules

The rules are as follows.

  1. Play begins with each player making an Ante wager. There is also an optional Progressive Jackpot side bet.
  2. The dealer gives the player and himself two hole cards and three community cards face up on the table.
  3. Each player must decide to either fold or call. If the player folds, then he gives up his cards and his Ante bet. If the player calls, then the Call bet must be equal to two times the Ante bet.
  4. The dealer will then deal two more community cards, for a total of five. The dealer will then turn over his own two cards.
  5. Both hands shall be scored according the highest poker value of the two hole cards and five community cards.
  6. The dealer must have a pair of fours or better to qualify. If the dealer does not qualify then the Ante will pay according to the Ante pay table below and the Call bet will push.
  7. If the dealer qualifies and beats the player, then the player will lose both the Ante and Call.
  8. If the dealer qualifies and the player beats the dealer then the Ante will pay according to the Ante pay table below and the Call bet will pay 1 to 1.
  9. If the dealer qualifies and the player ties the dealer then both Ante and Call bets will push.
  10. The Progressive Jackpot side bet pays based on the player's final seven card hand only, even if the player folded or loses to the dealer. More on this bet at the bottom of the page.

Ante Pay Table

Hand Table 1 Table 2 Table 3 Table 4
Royal flush 20 25 100 100
Straight flush 20 25 20 49
4 of a kind 10 12 10 17
Full house 3 3 3 3
Flush 2 2 2 2
All other 1 1 1 1

Analysis

The following table shows the number of combination, probability, and contribution to the return of all possible outcomes under ante pay table 3, which I'm told is the most frequently used. The lower right cell shows a house edge of 2.16%.

Return Table — Pay Table 3Expand

Hand Pays Combinations Probability Return
Player wins with royal flush 102 736259040 0.000026 0.002700
Player has royal flush, dealer doesn't qualify 100 119892960 0.000004 0.000431
Player wins with straight flush 22 6001122284 0.000216 0.004747
Player has straight flush, dealer doesn't qualify 20 1089672828 0.000039 0.000784
Player wins with four of a kind 12 41419896552 0.001489 0.01787
Player has four of a kind, dealer doesn't qualify 10 909499320 0.000033 0.000327
Player wins with full house 5 589505037660 0.021195 0.105973
Player wins with flush 4 505131419580 0.018161 0.072645
Player wins with straight or less 3 6426670016572 0.23106 0.693181
Player has full house, dealer doesn't qualify 3 24601676832 0.000885 0.002654
Player has flush, dealer doesn't qualify 2 168883536432 0.006072 0.012144
Player has straight or less, dealer doesn't qualify 1 5521376219148 0.198512 0.198512
Tie 0 678596916580 0.024398 0
Player folds -1 5009067102600 0.180093 -0.180093
Player loses -3 8839701755612 0.317817 -0.953451
Total 27813810024000 1 -0.021576

The following return table is four pay table 4. The lower right cell shows a house edge of 0.35%. This pay table can be found at Internet casinos using Gamesys N.V. software titled "Casino Hold 'Em with Zero House Edge." As you can see, the "Zero House Edge" part is a misnomer. The player must pay a 10% commission on any net gambling win per session. A "session" ends after a period of one hour with no bet or 24 hours, whichever happens first.

Return Table — Pay Table 4Expand

Hand Pays Combinations Probability Return
Player wins with royal flush 102 736,259,040 0.000026 0.002700
Player has royal flush, dealer doesn't qualify 100 119,892,960 0.000004 0.000431
Player wins with straight flush 51 6,031,161,932 0.000217 0.011059
Player has straight flush, dealer doesn't qualify 49 1,096,758,936 0.000039 0.001932
Player wins with four of a kind 19 41,421,337,056 0.001489 0.028295
Player has four of a kind, dealer doesn't qualify 17 909,634,104 0.000033 0.000556
Player wins with full house 5 589,638,417,816 0.021199 0.105997
Player wins with flush 4 505,727,272,920 0.018183 0.072730
Player wins with straight or less 3 6,429,871,396,516 0.231175 0.693526
Player has full house, dealer doesn't qualify 3 24,601,738,176 0.000885 0.002654
Player has flush, dealer doesn't qualify 2 169,051,416,912 0.006078 0.012156
Player has straight or less, dealer doesn't qualify 1 5,526,154,486,512 0.198684 0.198684
Tie 0 679,763,871,140 0.024440 0.000000
Player folds -1 4,984,375,678,920 0.179205 -0.179205
Player loses -3 8,854,310,701,060 0.318342 -0.955027
Total 27,813,810,024,000 1.000000 -0.003511

The only pay tables fully analyzed are 3 and 4, because it takes my computer about a month to cycle through all possible combinations. However, assuming the player was following optimal strategy under pay table 3 but playing under pay table 1 or 2, the house edge under pay table 1 would be 2.40%, and under pay table 2 would be 1.96%. The actual house edge under pay tables 1 and 2 would be slightly less or equal to these figures.

Strategy

Unfortunately, there is no easy way to quantify optimal strategy for this game. I can say that the optimal strategy player will raise 82% of the time. So only in the worst 18% of hands should the player fold. In general these are when the player has two singletons in the hole that are low compared to the flop, with little or no chance for a straight or flush.

Side Bets

The only side bet I am aware of for this game is known as the AA+. It pays based on the poker value of the player's hand after the flop. There are three known pay tables, as follows:

AA+ Pay Tables

Hand Pay Table 1 Pay Table 2 Pay Table 3
Royal Flush 25 100 100
Straight flush 25 50 50
4 of a kind 25 40 40
Full house 25 30 30
Flush 25 20 20
Straight 7 7 10
Three of a kind 7 7 8
Two pair 7 7 7
Pair of aces 7 7 7
House edge 6.40% 6.26% 2.97%

Pay tables 1 and 2 can be found in land casinos. Pay table 3 is used by Internet casinos using Gamesys software.

As an example, following is the full return table for Pay Table 1.

AA+ Side Bet — Pay Table 1

Hand Pays Combinations Probability Return
Royal Flush 25 4 0.000002 0.000038
Straight flush 25 36 0.000014 0.000346
4 of a kind 25 624 0.000240 0.006002
Full house 25 3744 0.001441 0.036014
Flush 25 5108 0.001965 0.049135
Straight 7 10200 0.003925 0.027473
Three of a kind 7 54912 0.021128 0.147899
Two pair 7 123552 0.047539 0.332773
Pair of aces 7 84480 0.032505 0.227537
Other -1 2316300 0.891241 -0.891241
Total 2598960 1 -0.064023

Progressive Jackpot

The Progressive Jackpot bet shall pay based on the player's two hole cards and the five community cards only. It does not matter whether or not the player folds or loses to the dealer. In the event of a straight flush or royal flush on the board, all players who made the Progressive Jackpot bet shall split the win.

The following table shows the probability and contribution to the return for each event. Obviously, the return from the top two awards will depend on the jackpot size, so are treated as unknowns.

Progressive Jackpot

Hand Pays Combinations Probability Return
Royal flush Jackpot 4,324 0.000032 x
Straight flush 10% of Jackpot 37,260 0.000279 y
4 of a kind $100.00 224,848 0.001681 0.168100
Full house $10.00 3,473,184 0.025961 0.259610
Other $0.00 130,044,944 0.972047 0.000000
Total $0.00 133,784,560 1.000000 0.487931+x+y

The table above shows the fixed wins return 42.77% of the amount bet. Assuming the player is playing alone, for every $1,000 in the meter the return will increase by 6.02%. Under the same assumption, the break-even point is $9,503.22. Due to jackpot sharing, the return will go down slightly according to the number of other players making the bet.

Jumbo Jackpot

There is another progressive jackpot called the "Jumbo Jackpot." The following table shows the wins, probabilities and the contribution to the return for each type of winning hand.

Jumbo Jackpot

Hand Pays Combinations Probability Return
7-card straight flush ? 32 0.00000024 x
6-card straight flush $5,000.00 752 0.00000562 0.028105
5-card straight flush $250.00 40,800 0.00030497 0.076242
4 of a kind $50.00 224,848 0.00168067 0.084034
Full house $5.00 3,473,184 0.02596102 0.129805
Flush $4.00 4,047,644 0.03025494 0.121020
Straight $2.00 6,180,020 0.04619382 0.092388
Other $0.00 119,817,280 0.89559871 0.000000
Total $0.00 133,784,560 1.00000000 0.531593

The return for the top award depends on the amount in the jackpot. I can say that all the other wins return 53.16% of money bet. For every 100,000 bet units in the meter the return increases by 2.39%. At a jackpot of $1,958,300.75 the return reaches 100%.

Methodology

The above analysis was performed using a brute force combinatorial program that analyzed all 27,813,810,024,000 possible outcomes, and played every hand according to optimal player strategy. It took my computers about a month to crank though the 27.8 trillion hands.