Last Updated: May 27, 2014

# 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

1. 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.
2. Player must make equal wagers on the Ante and Bonus.
4. The player may either (1) fold or (2) make a Draw bet and draw 0 to 3 new cards.
5. The dealer will take the discards and replace them with new cards.
6. The player may either (1) fold or (2) make a second Draw bet and draw 0 or 1 new cards.
7. If the player discards, then the dealer will take the discard and replace it with a new card.
8. If the player has at least a two pair then the Ante and Draw bets will pay even money, otherwise they will lose.
9. 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.

## 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.