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Riverboat Hold'em Poker

Introduction

Riverboat Hold'em Poker is a poker-based table game found in Mississippi and Indiana. Usually I do all my own analysis for this site. However in the case of this game the math is by Stanley Ko. The game owner provided me Stanley's report, with permission to publish on this site. I can only independently vouch for the River bet tables.

Rules

  1. A single 52-card deck is used.
  2. Play starts with the player making an ante bet and an optional River Bet.
  3. The dealer will give each player ahd himself two cards face down. The player may look at his own cards but sharing of information is not allowed.
  4. The player has the following options.
    • Fold: Player loses ante bet. The River Bet (if made) remains active.
    • Call: Player must place an additional bet equal exactly to the ante.
    • Raise: Player must place an additional bet equal exactly to two times the ante.
  5. The dealer deals three community cards face up.
  6. The dealer will turn over his two hole cards.
  7. The dealer must have a king-jack or higher to qualify. For example, the lowest possible qualifying hand would be K,J,4,3,2. If the dealer does not qualify all ante bets will pay 1 to 1 and all call and raise bets will push.
  8. If the dealer qualifies and the player has the higher hand then the ante and call or raise bets will pay 1 to 1.
  9. If the dealer qualifies and the dealer has the higher hand, or there is a tie, then the ante and call or raise bets will lose.
  10. The dealer will resolve the river bet based only on the poker value of the player's hand. There are four pay tables available from the game maker as shown below.

    11. Strategy

    The following table shows the optimal player strategy according to the player's initial two cards. There is no code for fold because the player should never do so.

    Analysis

    Following optimal strategy the house edge of the ante bet is 2.2240%. On average the player will call 73.3032% of the time, and raise 26.6968%, for an average final wager of 2.2670 units. Thus the element of risk is 0.9810%.

    The following table shows the probability of each initial hand the probability of each outcome given per hand.

    Probability Table for Riverboat Hold'em

    Initial Hand Probability Win 3 Win 2 Win 1 Lose 2 Lose 3
    Unsuited 23 0.00905 0 0.128983 0.219176 0.651841 0
    Unsuited 24 0.00905 0 0.136228 0.219296 0.644476 0
    Unsuited 25 0.00905 0 0.146682 0.219417 0.633901 0
    Unsuited 26 0.00905 0 0.157035 0.219537 0.623429 0
    Unsuited 27 0.00905 0 0.172098 0.219567 0.608335 0
    Unsuited 28 0.00905 0 0.192181 0.219567 0.588252 0
    Unsuited 29 0.00905 0 0.215257 0.219447 0.565296 0
    Unsuited 2T 0.00905 0 0.241543 0.219326 0.539131 0
    Unsuited 2J 0.00905 0 0.270936 0.225312 0.503753 0
    Unsuited 2Q 0.00905 0 0.30825 0.225191 0.466559 0
    Unsuited 2K 0.00905 0 0.332831 0.234335 0.432834 0
    Unsuited 2A 0.00905 0 0.372709 0.247808 0.379483 0
    Unsuited 34 0.00905 0 0.145283 0.219386 0.635331 0
    Unsuited 35 0.00905 0 0.155737 0.219507 0.624756 0
    Unsuited 36 0.00905 0 0.16609 0.219627 0.614283 0
    Unsuited 37 0.00905 0 0.181189 0.219657 0.599154 0
    Unsuited 38 0.00905 0 0.199426 0.219687 0.580887 0
    Unsuited 39 0.00905 0 0.222502 0.219567 0.557931 0
    Unsuited 3T 0.00905 0 0.248788 0.219447 0.531765 0
    Unsuited 3J 0.00905 0 0.278722 0.225432 0.495846 0
    Unsuited 3Q 0.00905 0 0.316037 0.225312 0.458652 0
    Unsuited 3K 0.00905 0 0.341375 0.234455 0.424169 0
    Unsuited 3A 0.00905 0.382518 0 0.247928 0 0.369554
    Unsuited 45 0.00905 0 0.164793 0.219597 0.61561 0
    Unsuited 46 0.00905 0 0.175145 0.219717 0.605138 0
    Unsuited 47 0.00905 0 0.190244 0.219747 0.590008 0
    Unsuited 48 0.00905 0 0.208517 0.219778 0.571705 0
    Unsuited 49 0.00905 0 0.229747 0.219687 0.550566 0
    Unsuited 4T 0.00905 0 0.256033 0.219567 0.5244 0
    Unsuited 4J 0.00905 0 0.286509 0.225552 0.487939 0
    Unsuited 4Q 0.00905 0 0.323823 0.225432 0.450745 0
    Unsuited 4K 0.00905 0 0.34992 0.234575 0.415504 0
    Unsuited 4A 0.00905 0.392326 0 0.248048 0 0.359625
    Unsuited 56 0.00905 0 0.1842 0.219807 0.595992 0
    Unsuited 57 0.00905 0 0.1993 0.219838 0.580863 0
    Unsuited 58 0.00905 0 0.217573 0.219868 0.56256 0
    Unsuited 59 0.00905 0 0.238838 0.219778 0.541384 0
    Unsuited 5T 0.00905 0 0.263278 0.219687 0.517035 0
    Unsuited 5J 0.00905 0 0.294295 0.225673 0.480033 0
    Unsuited 5Q 0.00905 0 0.33161 0.225552 0.442838 0
    Unsuited 5K 0.00905 0 0.358465 0.234696 0.406839 0
    Unsuited 5A 0.00905 0.402134 0 0.248169 0 0.349696
    Unsuited 67 0.00905 0 0.208391 0.219928 0.571681 0
    Unsuited 68 0.00905 0 0.226664 0.219958 0.553378 0
    Unsuited 69 0.00905 0 0.24793 0.219868 0.532202 0
    Unsuited 6T 0.00905 0 0.272405 0.219778 0.507817 0
    Unsuited 6J 0.00905 0 0.302118 0.225793 0.47209 0
    Unsuited 6Q 0.00905 0 0.339432 0.225673 0.434895 0
    Unsuited 6K 0.00905 0 0.367046 0.234816 0.398138 0
    Unsuited 6A 0.00905 0.410232 0 0.248289 0 0.341479
    Unsuited 78 0.00905 0 0.235984 0.219928 0.544088 0
    Unsuited 79 0.00905 0 0.25725 0.219838 0.522913 0
    Unsuited 7T 0.00905 0 0.281725 0.219747 0.498527 0
    Unsuited 7J 0.00905 0 0.311739 0.225763 0.462499 0
    Unsuited 7Q 0.00905 0 0.347219 0.225673 0.427109 0
    Unsuited 7K 0.00905 0 0.37559 0.234816 0.389593 0
    Unsuited 7A 0.00905 0.42004 0 0.248289 0 0.33167
    Unsuited 89 0.00905 0 0.266534 0.219807 0.513659 0
    Unsuited 8T 0.00905 0 0.291009 0.219717 0.489273 0
    Unsuited 8J 0.00905 0 0.321564 0.225733 0.452704 0
    Unsuited 8Q 0.00905 0 0.35708 0.225642 0.417278 0
    Unsuited 8K 0.00905 0.384135 0 0.234816 0 0.381049
    Unsuited 8A 0.00905 0.429849 0 0.248289 0 0.321862
    Unsuited 9T 0.00905 0 0.299989 0.219597 0.480414 0
    Unsuited 9J 0.00905 0 0.331037 0.225612 0.443351 0
    Unsuited 9Q 0.00905 0 0.366554 0.225522 0.407925 0
    Unsuited 9K 0.00905 0.394367 0 0.234696 0 0.370937
    Unsuited 9A 0.00905 0.439777 0 0.248169 0 0.312054
    Unsuited TJ 0.00905 0 0.340186 0.225492 0.434323 0
    Unsuited TQ 0.00905 0 0.375654 0.225402 0.398945 0
    Unsuited TK 0.00905 0.404225 0 0.234575 0 0.361199
    Unsuited TA 0.00905 0.450899 0 0.248048 0 0.301052
    Unsuited JQ 0.00905 0.385222 0 0.232021 0 0.382757
    Unsuited JK 0.00905 0.407468 0 0.239503 0 0.353028
    Unsuited JA 0.00905 0.453968 0 0.254668 0 0.291364
    Unsuited QK 0.00905 0.417325 0 0.239383 0 0.343292
    Unsuited QA 0.00905 0.463776 0 0.254547 0 0.281676
    Unsuited KA 0.00905 0.463523 0 0.264537 0 0.27194
    Suited 23 0.003017 0 0.135208 0.219127 0.645665 0
    Suited 24 0.003017 0 0.14242 0.219247 0.638333 0
    Suited 25 0.003017 0 0.152818 0.219367 0.627814 0
    Suited 26 0.003017 0 0.16314 0.219488 0.617372 0
    Suited 27 0.003017 0 0.178128 0.219518 0.602354 0
    Suited 28 0.003017 0 0.198089 0.219518 0.582393 0
    Suited 29 0.003017 0 0.221023 0.219397 0.559579 0
    Suited 2T 0.003017 0 0.247144 0.219277 0.533579 0
    Suited 2J 0.003017 0 0.276258 0.225272 0.49847 0
    Suited 2Q 0.003017 0 0.313295 0.225152 0.461553 0
    Suited 2K 0.003017 0 0.337669 0.234308 0.428023 0
    Suited 2A 0.003017 0.377063 0 0.247808 0 0.375129
    Suited 34 0.003017 0 0.151419 0.219338 0.629243 0
    Suited 35 0.003017 0 0.161817 0.219458 0.618725 0
    Suited 36 0.003017 0 0.172139 0.219578 0.608283 0
    Suited 37 0.003017 0 0.187163 0.219608 0.593229 0
    Suited 38 0.003017 0 0.205301 0.219638 0.57506 0
    Suited 39 0.003017 0 0.228236 0.219518 0.552246 0
    Suited 3T 0.003017 0 0.254357 0.219397 0.526246 0
    Suited 3J 0.003017 0 0.284004 0.225392 0.490604 0
    Suited 3Q 0.003017 0 0.321041 0.225272 0.453687 0
    Suited 3K 0.003017 0 0.346161 0.234429 0.419411 0
    Suited 3A 0.003017 0.386799 0 0.247928 0 0.365272
    Suited 45 0.003017 0 0.170816 0.219549 0.609635 0
    Suited 46 0.003017 0 0.181138 0.219669 0.599192 0
    Suited 47 0.003017 0 0.196162 0.219699 0.58414 0
    Suited 48 0.003017 0 0.214336 0.219729 0.565935 0
    Suited 49 0.003017 0 0.235449 0.219638 0.544913 0
    Suited 4T 0.003017 0 0.26157 0.219518 0.518912 0
    Suited 4J 0.003017 0 0.291749 0.225513 0.482738 0
    Suited 4Q 0.003017 0 0.328786 0.225392 0.445821 0
    Suited 4K 0.003017 0 0.354653 0.234549 0.410798 0
    Suited 4A 0.003017 0.396535 0 0.248048 0 0.355416
    Suited 56 0.003017 0 0.190137 0.21976 0.590103 0
    Suited 57 0.003017 0 0.205161 0.219789 0.575049 0
    Suited 58 0.003017 0 0.223335 0.219819 0.556846 0
    Suited 59 0.003017 0 0.244483 0.219729 0.535788 0
    Suited 5T 0.003017 0 0.268782 0.219638 0.511579 0
    Suited 5J 0.003017 0 0.299495 0.225633 0.474871 0
    Suited 5Q 0.003017 0 0.336532 0.225513 0.437955 0
    Suited 5K 0.003017 0 0.363145 0.234669 0.402186 0
    Suited 5A 0.003017 0.406271 0 0.248169 0 0.345559
    Suited 67 0.003017 0 0.214198 0.21988 0.565922 0
    Suited 68 0.003017 0 0.232372 0.21991 0.547718 0
    Suited 69 0.003017 0 0.253521 0.219819 0.526661 0
    Suited 6T 0.003017 0 0.277855 0.219729 0.502416 0
    Suited 6J 0.003017 0 0.307279 0.225753 0.466967 0
    Suited 6Q 0.003017 0 0.344316 0.225633 0.43005 0
    Suited 6K 0.003017 0 0.371676 0.23479 0.393535 0
    Suited 6A 0.003017 0.414323 0 0.248289 0 0.337387
    Suited 78 0.003017 0 0.241636 0.21988 0.538484 0
    Suited 79 0.003017 0 0.262784 0.219789 0.517426 0
    Suited 7T 0.003017 0 0.287119 0.219699 0.493183 0
    Suited 7J 0.003017 0 0.316839 0.225724 0.457437 0
    Suited 7Q 0.003017 0 0.352067 0.225633 0.4223 0
    Suited 7K 0.003017 0 0.380172 0.23479 0.385038 0
    Suited 7A 0.003017 0.424064 0 0.248289 0 0.327647
    Suited 89 0.003017 0 0.272012 0.21976 0.508228 0
    Suited 8T 0.003017 0 0.296347 0.219669 0.483984 0
    Suited 8J 0.003017 0 0.3266 0.225694 0.447706 0
    Suited 8Q 0.003017 0 0.361863 0.225603 0.412533 0
    Suited 8K 0.003017 0.388669 0 0.23479 0 0.376542
    Suited 8A 0.003017 0.433804 0 0.248289 0 0.317906
    Suited 9T 0.003017 0 0.305277 0.219549 0.475174 0
    Suited 9J 0.003017 0 0.336016 0.225573 0.438411 0
    Suited 9Q 0.003017 0 0.371279 0.225483 0.403238 0
    Suited 9K 0.003017 0.398831 0 0.234669 0 0.3665
    Suited 9A 0.003017 0.443665 0 0.248169 0 0.308166
    Suited TJ 0.003017 0 0.345111 0.225453 0.429436 0
    Suited TQ 0.003017 0 0.380327 0.225362 0.394311 0
    Suited TK 0.003017 0.408625 0 0.234549 0 0.356826
    Suited TA 0.003017 0.454703 0 0.248048 0 0.297248
    Suited JQ 0.003017 0.389739 0 0.231982 0 0.378279
    Suited JK 0.003017 0.411805 0 0.239503 0 0.348692
    Suited JA 0.003017 0.457708 0 0.254668 0 0.287624
    Suited QK 0.003017 0.421597 0 0.239383 0 0.33902
    Suited QA 0.003017 0.467453 0 0.254547 0 0.277999
    Suited KA 0.003017 0.467135 0 0.264537 0 0.268328
    Pair 22 0.004525 0.447155 0 0.215914 0 0.336931
    Pair 33 0.004525 0.471466 0 0.216155 0 0.312379
    Pair 44 0.004525 0.495777 0 0.216395 0 0.287827
    Pair 55 0.004525 0.520088 0 0.216636 0 0.263276
    Pair 66 0.004525 0.544158 0 0.216877 0 0.238965
    Pair 77 0.004525 0.568469 0 0.216877 0 0.214654
    Pair 88 0.004525 0.592779 0 0.216877 0 0.190343
    Pair 99 0.004525 0.617331 0 0.216636 0 0.166033
    Pair TT 0.004525 0.641882 0 0.216395 0 0.141722
    Pair JJ 0.004525 0.652714 0 0.229634 0 0.117652
    Pair QQ 0.004525 0.677024 0 0.229394 0 0.093582
    Pair KK 0.004525 0.681116 0 0.249372 0 0.069511
    Pair AA 0.004525 0.675098 0 0.279701 0 0.045201

    The next table shows the overall probability of each possible outcome.

    Return Table for Riverboat Hold'em

    Win Probability Return
    3 0.121363 0.364088
    2 0.192021 0.384042
    1 0.228022 0.228022
    -2 0.377389 -0.754779
    -3 0.081205 -0.243614
    Total 1 -0.02224

    River Bet

    Literature from the game maker contains four possible pay tables for the River bet. The four return tables below show the four pay tables, the probability and return of each outcome, and the player's expected value in the lower right cell.

    I am told by the game maker that the Mississippi casinos follow pay table 3, and the Indiana casinos follow pay table 4.

    River Bet - Pay Table 1

    Event Pays Combinations Probability Return
    Royal flush 10000 4 0.000002 0.015391
    Straight flush 1000 36 0.000014 0.013852
    Four of a kind 100 624 0.00024 0.02401
    Full house 20 3744 0.001441 0.028812
    Flush 15 5108 0.001965 0.029481
    Straight 8 10200 0.003925 0.031397
    Three of a kind 4 54912 0.021128 0.084514
    Two pair 3 123552 0.047539 0.142617
    Nines or better 1 506880 0.195032 0.195032
    All other -1 1893900 0.728715 -0.728715
    Total 2598960 1 -0.16361

    River Bet - Pay Table 2

    Event Pays Combinations Probability Return
    Royal flush 250 4 0.000002 0.000385
    Straight flush 50 36 0.000014 0.000693
    Four of a kind 20 624 0.00024 0.004802
    Full house 12 3744 0.001441 0.017287
    Flush 9 5108 0.001965 0.017689
    Straight 5 10200 0.003925 0.019623
    Three of a kind 4 54912 0.021128 0.084514
    Two pair 3 123552 0.047539 0.142617
    Nines or better 2 506880 0.195032 0.390064
    All other -1 1893900 0.728715 -0.728715
    Total 2598960 1 -0.051042

    River Bet - Pay Table 3 (Mississippi)

    Event Pays Combinations Probability Return
    Royal flush 250 4 0.000002 0.000385
    Straight flush 50 36 0.000014 0.000693
    Four of a kind 25 624 0.00024 0.006002
    Full house 15 3744 0.001441 0.021609
    Flush 10 5108 0.001965 0.019654
    Straight 8 10200 0.003925 0.031397
    Three of a kind 5 54912 0.021128 0.105642
    Two pair 3 123552 0.047539 0.142617
    Sevens or better 1 675840 0.260042 0.260042
    All other -1 1724940 0.663704 -0.663704
    Total 2598960 1 -0.075663

    River Bet - Pay Table 4 (Indiana)

    Event Pays Combinations Probability Return
    Royal flush 500 4 0.000002 0.00077
    Straight flush 100 36 0.000014 0.001385
    Four of a kind 40 624 0.00024 0.009604
    Full house 15 3744 0.001441 0.021609
    Flush 8 5108 0.001965 0.015723
    Straight 6 10200 0.003925 0.023548
    Three of a kind 4 54912 0.021128 0.084514
    Two pair 3 123552 0.047539 0.142617
    Sixes or better 1 760320 0.292548 0.292548
    All other -1 1640460 0.631199 -0.631199
    Total 2598960 1 -0.038882

Written by: Michael Shackleford