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Three Card Poker Ante and Play Analysis

3 Card Poker - Ante and Play Analysis

The analysis of the Ante and Play bet found in Three Card Poker can be divided up into two seperate parts: a bonus component and a non-bonus component. The bonus component consists of the Ante Bet Bonus which is awarded independently of the dealer's hand or whether the player wins, loses or ties. The non-bonus component consists of the Ante bet and whether the player decides to make the Play bet or not after seeing their hand.

All results in this analysis are given in terms of the initial ante bet of 1 unit as opposed to the total amount bet by the player (i.e. Ante plus Play bet if made). Therefore a house advantage of x% represents a player losing $x per every $100 bet on the Ante bet.

Non-Bonus Analysis

The non-bonus analysis was done using a program written in C++ that exhaustively analyzes all possible combinations of player and dealer hands. It is possible to reduce the combinations examined by using suit symmetries (e.g. AHeart ADiamond KDiamond is equivalent to AClub ASpadeKSpade) but was not necessary as the brute-force method runs in a reasonable amount of time (~15 sec on 3GHz Pentium).

Number of Combinations
The total number of combinations to determine the house advantage for Three Card Poker is selecting 3 cards from a standard 52 card deck for the player and 3 cards from the remaining 49 cards for the dealer.

Total Number of Combinations = 52C3 x 49C3 = 22100 x 18424 = 407170400

Optimal Strategy
The optimal strategy for the player assuming knowledge of their own 3 card hand only is to make the Play bet holding Q-6-4 High or better.

Results
Results were calculated for the optimal player strategy in addition to other player strategies. These results tend to indicate it is a greater error to fold too often as compared to making the Play bet too often.

Player StrategyNon-Bonus Return
Play Any Hand-0.129389
Play J High or better-0.100696
Play Q High or better-0.087342
Play Q-6-4 High or better * -0.086580
Play K High or better-0.102706
Play A High or better-0.203344
Play Any Pair or better-0.441216

* Optimal Strategy

Bonus Analysis

Using the Ante Bet Bonus pay table listed in the Standard Rules the player gains 0.052851 of an ante bet as calculated below. For different pay tables use the Ante and Play Odds Calculator to determine the house advantage.

HandFrequency Probability Bonus Pay OutReturn
Straight Flush480.0021725 to 10.010860
Three of a Kind520.0023534 to 10.009412
Straight7200.0325791 to 10.032579
Any Other212800.9628960 to 10.000000
Total221001.000000 0.052851

Total House Advantage

The total house advantage if the player uses optimal strategy is 3.37%. Note that the house advantage for the Ante Bet Bonus is negative as it operates in favor of the player.

Player StrategyNon-Bonus ReturnBonus ReturnHouse Advantage
Play Any Hand12.938862%-5.285068%7.653794%
Play J High or better10.069648%-5.285068%4.784581%
Play Q High or better8.734189%-5.285068%3.449121%
Play Q-6-4 High or better *8.658049%-5.285068%3.372981%
Play K High or better10.270601%-5.285068%4.985533%
Play A High or better20.334414%-5.285068%15.049346%
Play Any Pair or better44.121552%-5.285068%38.836484%

* Optimal Strategy

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