Royal Draw

Introduction

Royal Draw is a video poker variation by Bally. I noticed it at the Suncoast in Las Vegas on November 7, 2006. The rules are the same as conventional video poker except the player plays the Royal Draw bonus with any four of a kind, in addition to his regular four of a kind win. The bonus round is like another hand of video poker except with a 13-card deck, one each of the 13 cards of a suit chosen by the player. The pay table in the bonus round is the same as the base game, based on the same number of coins bet as the initiating game. Comparing the "9/6" Royal Draw game to "9/6" Jacks or Better, the former will have 3.317 times as many royals.

Proper strategy in the bonus round is to go for the royal, with the exception to always keep a pat straight flush. Following optimal strategy, the return table of the bonus round is as follows.

Royal Draw Bonus Round Return Table

Hand Payoff Combinations Probability Return
Royal flush 800 436939 0.024745 19.796025
Straight flush 50 282877 0.01602 0.801004
Flush 6 16937830 0.959235 5.755409
Total 17657646 1 26.352438

The lower right cell shows the bonus has an average value of 26.352438 times the amount bet. Adding that to the value of a four of a kind (50) results in an average win of 76.352438 for a four of a kind. The following is the pay table for the "9/6" game I noticed at the Suncoast, substituting the average value for a four of a kind.

Royal Draw "9/6" Pay Table

Hand Payoff Combinations Return
Royal flush 800 473481924 0.000024 0.019003
Straight Flush 50 2172269688 0.000109 0.005449
Four of a Kind + Bonus 76.352438 47014134108 0.002359 0.180083
Full House 9 228947750892 0.011486 0.103372
Flush 6 221544090528 0.011114 0.066686
Straight 4 255600824868 0.012823 0.051291
Three of a kind 3 1476003456204 0.074047 0.222142
Two pair 1 2558163543084 0.128337 0.128337
Pair 1 4214845162188 0.211448 0.211448
Nonpaying hand 0 10928465803716 0.548254 0
Total 19933230517200 1 0.987811

Here is the "9/6" pay table, breaking down the possible Royal Draw bonus events. Combinations are to 15 significant digits (a constraint I hate in Excel).

Royal Draw "9/6" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 8,360,576,201,390,900 0.000024 0.019003
Straight Flush 50 38,357,169,167,234,400 0.000109 0.005449
Four of a kind + royal flush 850 20,542,308,743,015,400 0.000058 0.049609
Four of a kind + straight flush 100 13,299,217,214,068,700 0.000038 0.003778
Four of a kind + flush 56 796,317,411,118,506,000 0.002262 0.126696
Full House 9 4,042,678,337,747,120,000 0.011486 0.103372
Flush 6 3,911,947,123,935,380,000 0.011114 0.066686
Straight 4 4,513,308,882,827,140,000 0.012823 0.051291
Three of a kind 3 26,062,746,524,426,700,000 0.074047 0.222142
Two pair 1 45,171,146,253,883,000,000 0.128337 0.128337
Pair 1 74,424,243,818,728,300,000 0.211448 0.211448
Nonpaying hand 0 192,970,980,485,123,000,000 0.548254 0
Total 351,973,928,109,115,000,000 1 0.987811

The rest are hypothetical pay tables, which may or may not ever exist in a real casino.

Royal Draw "6/5" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 491939760 0.000025 0.019744
Straight Flush 50 2157022404 0.000108 0.005411
Four of a Kind + Bonus 76.352438 49643321292 0.00249 0.190154
Full House 6 163845131004 0.00822 0.049318
Flush 5 216560104836 0.010864 0.054321
Straight 4 260290671468 0.013058 0.052233
Three of a kind 3 1559213724564 0.078222 0.234665
Two pair 1 2172303530532 0.108979 0.108979
Pair 1 4578690274164 0.229701 0.229701
Nonpaying hand 0 10930034797176 0.548332 0
Total 19933230517200 1 0.944526

Royal Draw "7/5" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 491854620 0.000025 0.01974
Straight Flush 50 2154468204 0.000108 0.005404
Four of a Kind + Bonus 76.352438 47005734660 0.002358 0.180051
Full House 7 228890267460 0.011483 0.08038
Flush 5 216500514576 0.010861 0.054306
Straight 4 260261261016 0.013057 0.052227
Three of a kind 3 1475241247320 0.074009 0.222027
Two pair 1 2556431280000 0.12825 0.12825
Pair 1 4216659716952 0.211539 0.211539
Nonpaying hand 0 10929594172392 0.54831 0
Total 19933230517200 1 0.953925

Royal Draw "8/5" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 491854620 0.000025 0.01974
Straight Flush 50 2154468204 0.000108 0.005404
Four of a Kind + Bonus 76.352438 47005734660 0.002358 0.180051
Full House 8 228890267460 0.011483 0.091863
Flush 5 216500514576 0.010861 0.054306
Straight 4 260261261016 0.013057 0.052227
Three of a kind 3 1475241247320 0.074009 0.222027
Two pair 1 2556431280000 0.12825 0.12825
Pair 1 4216659716952 0.211539 0.211539
Nonpaying hand 0 10929594172392 0.54831 0
Total 19933230517200 1 0.965408

Royal Draw "9/5" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 491856684 0.000025 0.01974
Straight Flush 50 2153793276 0.000108 0.005403
Four of a Kind + Bonus 76.352438 47005841988 0.002358 0.180052
Full House 9 228890861892 0.011483 0.103346
Flush 5 216486883920 0.010861 0.054303
Straight 4 260255023608 0.013056 0.052225
Three of a kind 3 1475246648808 0.074009 0.222028
Two pair 1 2556440400816 0.12825 0.12825
Pair 1 4216746386376 0.211544 0.211544
Nonpaying hand 0 10929512819832 0.548306 0
Total 19933230517200 1 0.976891

Royal Draw "8/6" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 473481924 0.000024 0.019003
Straight Flush 50 2172269688 0.000109 0.005449
Four of a Kind + Bonus 76.352438 47014134108 0.002359 0.180083
Full House 8 228947750892 0.011486 0.091886
Flush 6 221544090528 0.011114 0.066686
Straight 4 255600824868 0.012823 0.051291
Three of a kind 3 1476003456204 0.074047 0.222142
Two pair 1 2558163543084 0.128337 0.128337
Pair 1 4214845162188 0.211448 0.211448
Nonpaying hand 0 10928465803716 0.548254 0
Total 19933230517200 1 0.976325

Royal Draw "10/6" Pay Table

Hand Payoff Combinations Probability Return
Royal flush 800 473734764 0.000024 0.019013
Straight Flush 50 2173028208 0.000109 0.005451
Four of a Kind + Bonus 76.352438 47036533668 0.00236 0.180169
Full House 10 229024825812 0.01149 0.114896
Flush 6 221668234968 0.011121 0.066723
Straight 4 254739744708 0.01278 0.051119
Three of a kind 3 1476869429076 0.074091 0.222272
Two pair 1 2559283090740 0.128393 0.128393
Pair 1 4211166789108 0.211264 0.211264
Nonpaying hand 0 10930795106148 0.54837 0
Total 19933230517200 1 0.9993

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