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Double Draw Poker
Introduction
Double Draw Poker is a poker variant in which the player gets two draw opportunities to improve his hand. There is no dealer hand to beat, but the player must get at least a two pair to win.
Rules
- The game is played with a 54-card deck, including two jokers. The jokers are treated as in pai gow poker, where they can be used to complete a straight or flush. Otherwise, they are counted as an ace.
- Player must make equal wagers on the Ante and Bonus.
- Player receives five cards.
- The player may either (1) fold or (2) make a Draw bet and draw 0 to 3 new cards.
- The dealer will take the discards and replace them with new cards.
- The player may either (1) fold or (2) make a second Draw bet and draw 0 or 1 new cards.
- If the player discards, then the dealer will take the discard and replace it with a new card.
- If the player has at least a two pair then the Ante and Draw bets will pay even money, otherwise they will lose.
- The Bonus will pay according to the pay table below.
Following is the pay table. There are two known pay tables for the game. The first was seen at the Cromwell in Las Vegas. The second is reported to have been seen in Atlantic City somewhere. Note the only difference is in the three of a kind.
Bonus Bet Pay Table
| Hand | Cromwell | Atlantic City |
|---|---|---|
| Five of a Kind | 500 | 500 |
| Natural Royal Flush | 100 | 100 |
| Wild Royal Flush | 50 | 50 |
| Straight Flush | 50 | 50 |
| Four of a Kind | 20 | 20 |
| Full House | 5 | 5 |
| Flush | 3 | 3 |
| Straight | 2 | 2 |
| Three of a kind | 1 | Push |
| Two pair | Push | Push |
| Pair or less | Loss | Loss |
Analysis
The following return table shows the probability and return for all possible outcomes under the Cromwell pay table. The lower right cell shows a house edge of 6.72%.
Cromwell Return Table
| Event | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Win with Five Aces | 503 | 52,322,119,210,368 | 0.000118 | 0.059218 |
| Win with Natural Royal Flush | 103 | 14,993,158,995,360 | 0.000034 | 0.003475 |
| Win with Wild Royal Flush | 53 | 271,879,037,718,624 | 0.000612 | 0.032423 |
| Win with Straight Flush | 53 | 1,775,636,415,651,168 | 0.003995 | 0.211753 |
| Win with Four of a Kind | 23 | 2,560,394,313,544,032 | 0.005761 | 0.132506 |
| Win with Full House | 8 | 13,626,649,916,337,024 | 0.030661 | 0.245290 |
| Win with Flush | 6 | 24,666,264,348,582,528 | 0.055501 | 0.333008 |
| Win with Straight | 5 | 27,344,571,829,162,560 | 0.061528 | 0.307639 |
| Win with Three of a Kind | 4 | 36,005,432,366,344,896 | 0.081015 | 0.324062 |
| Win with Two Pair | 3 | 75,196,194,892,068,096 | 0.169198 | 0.507595 |
| Fold Immediately | -2 | 8,853,372,565,632,000 | 0.019921 | -0.039842 |
| Fold After First Draw | -3 | 45,484,750,646,995,968 | 0.102345 | -0.307034 |
| Draw Twice and Lose | -4 | 208,574,193,509,390,016 | 0.469311 | -1.877243 |
| Total | 444,426,655,119,632,640 | 1.000000 | -0.067152 |
The 6.72% house edge for the Cromwell pay table is the ratio of the expected loss to the Ante wager. While this is a good way to estimate expected losses over time, it is not a good metric of the value of the game compared to other games. On average, the player will bet 3.857813 by the end of the hand, including the Bonus and Draw bets. For comparing one game to another, I like to use the Element of Risk, which is the expected loss to the total wager. In this case, that is 6.72%/3.86 = 1.74%.
The following return table shows the probability and return for all possible outcomes under the Atlantic City pay table. The lower right cell shows a house edge of 14.77%.
Atlantic City Return Table
| Event | Pays | Combinations | Probability | Return |
|---|---|---|---|---|
| Win with Five Aces | 503 | 51,904,915,542,144 | 0.000117 | 0.058746 |
| Win with Natural Royal Flush | 103 | 14,995,776,898,368 | 0.000034 | 0.003475 |
| Win with Wild Royal Flush | 53 | 266,484,816,402,336 | 0.000600 | 0.031780 |
| Win with Straight Flush | 53 | 1,776,016,073,270,880 | 0.003996 | 0.211798 |
| Win with Four of a Kind | 23 | 2,532,689,494,376,832 | 0.005699 | 0.131072 |
| Win with Full House | 8 | 13,518,660,086,749,440 | 0.030418 | 0.243346 |
| Win with Flush | 6 | 24,944,894,229,894,912 | 0.056128 | 0.336770 |
| Win with Straight | 5 | 27,684,270,412,104,096 | 0.062292 | 0.311461 |
| Win with Three of a Kind | 3 | 35,526,730,871,316,384 | 0.079938 | 0.239815 |
| Win with Two Pair | 3 | 74,313,369,865,302,624 | 0.167212 | 0.501635 |
| Fold immediately | -2 | 12,775,838,201,376,768 | 0.028747 | -0.057494 |
| Fold after first draw | -3 | 44,073,443,129,558,976 | 0.099169 | -0.297508 |
| Place both bets but lose | -4 | 206,947,357,246,838,880 | 0.465650 | -1.862601 |
| Total | 444,426,655,119,632,640 | 1.000000 | -0.147705 |
The average bet for the Atlantic City paytable is 3.843337, and the Element of Risk is 3.84%.
Strategy
From the table above you can see that under the Cromwell rules the player will fold immediately 2.0% of the time and after the first draw 10.2% of the time. If that isn't enough for you, then please see my 5-page Double Draw Poker strategy (PDF). It shows how to play most situations in the first draw correctly. The second draw is usually obvious.
Rack Card
Acknowledgment
An analysis of this game requires looping through 28,587,727,358,190 different hands. To put that in comparison, that is seven times as many as video poker. The problem isn't writing a program to do it, but making it finish within months, or years.
When I threw my hands up in the air and said "forget it," my sidekick JB wasn't discouraged. Between his ingenious short cuts and multi-core processing, he managed to cut the processing time to several hours. So, all due credit to JB for the return tables and the strategies.
External Links
There are two threads about this game at my forum Wizard of Vegas:- Double Draw Poker (started May 3, 2014)
- Double Draw Poker (started May 28, 2013)