Statistical Analysis
Professional Opinion Assessing Poker Game 211 by Precision Consulting CompanyThis paper serves as a certified professional opinion of the trademarked card game “211.”
The paper will discuss and highlight all of the following aspects of this new card game:
(a) a brief discussion of the pros and cons of the rules and a perspective and analysis of the hi/lo aspect of the game;
(b) a shorter perspective and analysis of the hi only aspect of the game; and
(c) an overall conclusion regarding the marketability and attractiveness
of the card game to the casino going public in addition to the more casual card player.
Below, a table is presented that shows how many of each type of poker hand can be deal in 211
Hand

Number

Probability

Quads

13

0.0050%

Straight Flush

44

0.0160%

3ofakind

2496

0.9200%

Two pairs

2808

1.0200%

Flush

2,816

1.0400%

Pair

82,368

30.4200%

No Pair

177,408

65.5300%

These numbers were calculated combinatorically using logic and the choose operator. That same methodology applies to all calculations done for this opinion. Moving from hole cards to the board, there are two driving significant differences between Omaha and Holdem on the one hand and 211 on the other. Taking the first difference, in the former two games, there are 5 cards on the board yielding 2,598,960 possible boards. In 211, there are only 270,725 board possibilities.
Despite that difference, the total number of ways in which the cards from the board can be used are quite similar. In 211, a player can use either 3 cards from his or her hand and 2 from the board or the reciprocal. In Hold’em and Omaha, 3 cards from the board must be used while 2 from the hand must be used. As a result, there are 10 ways that 3 cards can be chosen from 5. Similarly, there are 6 ways that 3 cards can be chosen from 4 and 4 ways that 2 cards can be chosen from the 4 board cards. Therefore, there are the same number of cards from the board that can be used. The ranking from 211 to Hold’em with Omaha in the middle of how many ways the hole cards can be chosen is 10 for 211 (choosing 2 cards from 4 or 3 cards from 4), 6 for Omaha (choosing 2 cards from 4), and 2 for Holdem.
Therefore, from a combinatoric perspective, the card game of 211 is more complex from the hole card choosing perspective. The next step is determining the ratio of low card boards to high card boards. As opposed to Omaha, 211 does not use an 8 qualifier as the low. Instead, given the fewer number of board cards and the ability to use either 2 or 3 cards from the 4 hole cards, 211 uses a 7 low qualifier. Due to the combination of those two factors, there appears to be a higher probability of a low board available for use with 211 than with Omaha. The below table lists the relative probabilities of the appearance of usable low card boards for Omaha and 211. Specifically, the bolded numbers indicate boards that do not automatically preclude the use of a low:
Omaha

211

This table shows the number of boards that can exist with the condition of high cards being assessed first followed by low cards. One aspect that this table neglects is the issue of pairs. Low boards cannot exist if a pair is present with the low cards (or if 2 of the 4 low cards are paired in Omaha etc.). Since pair probabilities are proportional to the cards available (or three of a kind or four of a kind), both Omaha and 211 would be penalized by the pair (or 3 of a kind or 4 of a kind) probabilities proportionally. Therefore, the true probability of a board being accessible to a low hand would be around 60% for Omaha and around 63% for 211.
The final point of interest with respect to the rules of the hi/lo game of 211 (and the more general 211 game structure) involves the evolution of the betting rounds. The structure of the game is such that there are still 7 cards, similar to Hold’em, and to Omaha in the following way. The available cards for use to the player appear in the same basic order after the second round. For instance, there are 2, 3, and 4 visible cards in Hold’em, Omaha, and 211 respectively on the first round. After the flop, there are 5, 5, and 5 cards available with respect to the total number of cards that can be used to make a poker hand. The remaining rounds evolve identically with one card being added to the set of community cards each round. The most interesting aspect of the game appears to be that the probability of hitting individual high hands after the flop, from the prospective of the analyst or player, does not change in either game. The cards are revealed one at a time with a specific number of unknown cards relative to “known” cards. The 7low qualifier for the hi/lo aspect of the game alters the probability of the low being completed given the flop. There are fewer low cards that can come, however, the choice to use three low cards in the hole has, as demonstrated above, a significant offsetting effect on that probability.
The other aspect of the evolution involves the interaction of the players throughout the game. In this respect, the game is more similar to Omaha than Hold’em since low drawing hands are more likely to be out. The term for this is bunching. Specifically, if nx players, where n>x, fold their initial 4 card hand, the probability of Aces and 2s being in the remaining x hands increases. This increases nonlinearly as x increases. Hold’em does not exhibit this bunching effect. As a result, the perceptions of knowledgeable players regarding the low card evolution of the board given the action, evolves as it would in Omaha moreso than Hold’em. The evolution also includes the betting interaction and the thought processes of the players. Despite the variable number of cards used and known, and the structure of the game, the bluffing frequencies from a game theoretic perspective are virtually identical to how they would be calculated in Omaha. The difference simply revolves around the probability of having a low and the ways that the final card can help or hurt the opponents’ hands. The probabilities from the board perspective, given above, are slightly different than from the players’ perspectives. There are more ways that a player can be helped by a single card in 211 due to the choice of whether the hand has been made, could be made, or could be improved due to the variable number of hold cards and board cards that can be utilized. Since a higher proportion of help can be given with a card, semibluffing and bluffing must necessarily be higher from a game theoretic perspective. The next section details the highonly rule derivation.
211 High only:
The combination of rules that relate to high only do not seem to provide the same enthusiasm from the perspective of the player than the more dynamic hi/lo rules. The reason for this is very similar to why Omaha highonly has deteriorated in recent years and is only played in pot limit form. Highonly 211 in pot limit form can certainly compete with highonly pot limit Omaha since the games are so similar. Overall, as combinatorically shown above, the choice of 2 or 3 cards from the 4 card board does much offset the reduced number of board cards and makes the game equally, if not more, interesting. Overall, the opinion of this paper is such that the high only aspect of the game should only be spread in pot limit form. The hi/lo variation is more complex, though is not as amenable to pot limit play.
Marketability Assessment:
The main driver of a game’s ability to be spread is the demand of that game from the card playing public. The demand of the card playing public must begin at some point. There is no reason, or more specifically, no barrier to the spreading of 211 if the demand for the game increases to a point where the number of requests a casino gets for the game lead to the marginal Hold’em or Omaha table adding less to the satisfaction of demand than the next 211 game. The playing of the game, as was experimented and assessed, involves a similar degree of luck and identical enjoyment. It does take a small degree of accommodation in order to become comfortable with assessing the possibilities, however, once the counting becomes rote, the odds calculations are identical and the river play similarly, or slightly more complex than Omaha. As time progresses, players are likely to crave new variations of games in order to derive a similar level of enjoyment from playing. Further, 211 can be implemented fairly easily since the hole card dealing is the same as Omaha and the only dealing difference is the number of cards that appear on the flop. Overall, 211 possesses the right mix of luck and skill, the familiarity with rules, odds, and hand assessment, such that it can easily be adopted and only needs a sufficient backing in order to become a successful and powerful addition to the specter of the card playing public.