3 Card Poker - Ante and Play Strategy
Calculating Strategy
To determine the correct strategy for any hand of Three Card Poker it is necessary to calculate the expected value (E.V.) for each possible decision, in this case either to:
The correct strategy is to make which ever decision has the highest expected value.
For the purpose of these calculations the conditions are:
- Use the Standard Rules of Three Card Poker.
- The player has no information other than the three cards dealt to their hand.
- Expected value is calculated on the basis of an Ante bet of 1 unit as opposed to the total amount bet (i.e. Ante + Play bet).
- The Ante Bonus is ignored as it is awarded independently of the result of the hand.
The expected value of the player folding is simply equal to losing the ante bet which is -1.
The expected value of the player making the Play bet is calculated by cycling through all possible dealer hands (49C3 = 18424 possible dealer hands) and determining the frequency of each possible result. The table below illustrates the calculations involved in determining the expected value for making the Play bet given the frequency of each result. While it might be possible to do this calculation by hand, computers can easily and quickly calculate the frequencies and expected value.
| Result | Frequency | Probability | Win/Loss | Expected
Value |
|---|
| Dealer does not qualify | f1 | p1 = f1/18424 | 1 | EV1 = 1 x p1 |
| Dealer qualifies, player's hand wins | f2 | p2 = f2/18424 | 2 | EV2 = 2 x p2 |
| Dealer qualifies, player's hand ties | f3 | p3 = f3/18424 | 0 | 0 |
| Dealer qualifies, player's hand lose | f4 | p4 = f4/18424 | -2 | EV4 = -2 x p4 |
| Total | 18424 | 1.00000000 | | EV = EV1+EV2+EV4 |
Optimal Strategy
By calculating the expected value of each possible player hand it is determined that the optimal strategy is for the player to make the Play bet on all hands ranked Q-6-4 High or better. Any other hand should be folded. Below are the expected values for making the Play bet for Q-6-3 High, Q-6-4 High, and Q-6-5 High. Note that there are only 4 distinct suit combinations with other suit combinations being equivalent to the ones listed (e.g. Q 6 4 is equivalent to Q 6 4 ).
| Hand | No Qualify | Win | Tie | Lose | Play E.V. |
|---|
| Q 6 3 | 5747 | 271 | 26 | 12380 | -1.002551 |
| Q 6 3 | 5747 | 270 | 25 | 12382 | -1.002877 |
| Q 6 3 | 5747 | 268 | 25 | 12384 | -1.003311 |
| Q 6 3 | 5739 | 271 | 25 | 12389 | -1.003962 |
| Hand | No Qualify | Win | Tie | Lose | Play E.V. |
|---|
| Q 6 4 | 5758 | 305 | 26 | 12335 | -0.993378 |
| Q 6 4 | 5758 | 303 | 25 | 12338 | -0.993921 |
| Q 6 4 | 5758 | 302 | 25 | 12339 | -0.994138 |
| Q 6 4 | 5751 | 305 | 25 | 12343 | -0.994627 |
| Hand | No Qualify | Win | Tie | Lose | Play E.V. |
|---|
| Q 6 5 | 5754 | 339 | 26 | 12305 | -0.986648 |
| Q 6 5 | 5754 | 336 | 25 | 12309 | -0.987408 |
| Q 6 5 | 5754 | 336 | 25 | 12309 | -0.987408 |
| Q 6 5 | 5748 | 339 | 25 | 12312 | -0.987733 |
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