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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. A
A
K is equivalent to A
A
K) 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 Strategy | Non-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.
| Hand | Frequency | Probability | Bonus Pay Out | Return |
|---|---|---|---|---|
| Straight Flush | 48 | 0.002172 | 5 to 1 | 0.010860 |
| Three of a Kind | 52 | 0.002353 | 4 to 1 | 0.009412 |
| Straight | 720 | 0.032579 | 1 to 1 | 0.032579 |
| Any Other | 21280 | 0.962896 | 0 to 1 | 0.000000 |
| Total | 22100 | 1.000000 | 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 Strategy | Non-Bonus Return | Bonus Return | House Advantage |
|---|---|---|---|
| Play Any Hand | 12.938862% | -5.285068% | 7.653794% |
| Play J High or better | 10.069648% | -5.285068% | 4.784581% |
| Play Q High or better | 8.734189% | -5.285068% | 3.449121% |
| Play Q-6-4 High or better * | 8.658049% | -5.285068% | 3.372981% |
| Play K High or better | 10.270601% | -5.285068% | 4.985533% |
| Play A High or better | 20.334414% | -5.285068% | 15.049346% |
| Play Any Pair or better | 44.121552% | -5.285068% | 38.836484% |
* Optimal Strategy