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Hold 'Em Challenge
Introduction
Rules
Hold 'em Challenge is a poker based video game I have seen at the Mirage and Caesars Palace. Following are the rules.
- Game is played with a single 52-card deck.
- After player makes a wager and presses the deal button three 2-card hands are dealt face up.
- Player must choose one of the three hands to play. This will constitute the first two cards of the player's hand and the other two pairs will each count as the first cards of two separate opponents.
- Five more community cards are dealt face up.
- The game shall score the player's best 5-card poker hand using any combination of the player's 2 cards and the 5 community cards. Likewise the game will determine the best poker hand of the two opponents.
- The player shall be paid according to the outcome of his hand and the pay table below.
The following pay table is the only one I know of. All pays are on a "for one" basis, in other words the player never gets his original bet back, even if he wins.
Hold 'em Challenge Pay Table
| Event | Pays |
|---|---|
| On board royal flush | 2000 |
| Royal flush | 100 |
| Straight flush | 25 |
| Four of a kind | 10 |
| Full house | 3 |
| Flush | 3 |
| Straight | 3 |
| Three of a kind | 2 |
| Two pair | 2 |
| Jacks or better | 2 |
| Low pair | 1 |
| Garbage | 1 |
| Tie | 1 |
| Loss | 0 |
Return
The following table shows how often each event occurred in a random simulation. The lower right cell shows a return of 98.61%, or a house edge of 1.39%.
Hold 'em Challenge Return Table
| Event | Pays | Observations | Probability | Return |
|---|---|---|---|---|
| Royal on board | 2000 | 172295 | 0.000002 | 0.003071 |
| Royal Flush | 100 | 7345422 | 0.000065 | 0.006546 |
| Straight Flush | 25 | 28941374 | 0.000258 | 0.006448 |
| Four of a kind | 10 | 244685854 | 0.00218 | 0.021804 |
| Full House | 3 | 3024857400 | 0.026955 | 0.080865 |
| Flush | 3 | 3143432932 | 0.028012 | 0.084035 |
| Straight | 3 | 3604889822 | 0.032124 | 0.096372 |
| Three of a kind | 2 | 4541020679 | 0.040466 | 0.080932 |
| Two pair | 2 | 16958238827 | 0.151119 | 0.302237 |
| Pair: J-A | 2 | 10519745821 | 0.093744 | 0.187487 |
| Pair: 2-10 | 1 | 8297624269 | 0.073942 | 0.073942 |
| Ace high or less | 1 | 1805881455 | 0.016093 | 0.016093 |
| Tie | 1 | 2950191571 | 0.02629 | 0.02629 |
| Loss | 0 | 57091119243 | 0.508751 | 0 |
| Total | 112218146964 | 1 | 0.986122 |
Strategy
It isn't for lack of trying but I never quantified a good strategy for this game. A strategy based mostly on the strength of each 2-card combination was about 2% short of optimal. I speculate the major reason is the importance of penalty cards. So I'm afraid you are on your own with this one.
Methodology
The number of combinations in this game is so vast (418,597,840,861,200) it would take my computer about 220 years to do a perfect analysis. So I did a random simulation of 81,866 initial hands and 112,218,146,964 final hands. For each initial hand the program played out all combin(46,5) = 1370754 combinations of community cards and chose the hand with the greatest expected return. This simulation took 22.3 seconds per initial hand for a total of 21 days of computer time.