Three Card Poker

March 8, 2008

Three card poker is two games in one where the player may bet on either one, both, and in different amounts. Both games are based on hands consisting of three cards. Before I go on here are the possible hands in three card poker, the number of combinations of each hand, and the probability of forming. Note that a straight is harder to form than a flush.

Probabilities in Three Card Poker

PairplusThis is a simple game in which you get three cards and are paid according to their value. The dealer’s hand is immaterial. There is no raising and no discarding, in fact no skill of any kind is involved.The following table shows various pay tables for Pairplus. It used to be that the only table was pay table 1. Now pay table 4 is becoming increasingly prevalent in Las Vegas, especially on the Strip.

Payoff Table for Pairplus

Ante and Play

Play begins with a wager on ante. After the player views his three cards he may either raise by putting an equal bet on play or fold and lose the ante bet. If the player folds he also loses the pairplus bet if one was made, however this should not be any sacrifice because if the pairplus bet paid anything the player shouldn’t fold.If the player does raise then he goes against the dealer’s hand. The dealer needs at least a queen high to qualify. Below are the possible outcomes and their payoff:

Dealer does not qualify: Ante wins 1 to 1, play bet is returned
Dealer qualifies and player beats dealer: Both play and ante win 1 to 1
Dealer qualifies and dealer beats player: Both Play and ante lose
Dealer qualifies and dealer ties player: Both Play and ante push
In addition the Ante bet has an extra bonus that does not depend on the dealer’s hand. There are different variations of the bonus paytable. The following table shows some variations I have seen along with the corresponding game house edge and element of risk.

Ante Bet Bonus

Optimal strategy in ante and play is to raise if you have a queen/6/4 (that is a queen, 6, and 4 all in the same hand) or greater, regardless of the bonus pay table. Overall the player stands to lose 8.66% of the original wager but win 5.29% on the bonus.Many people have asked me what I mean by queen/6/4, wondering for example whether queen/7/3 is greater than queen/6/4. In any poker based game hands are scored first according to the highest card, then the second, and then the third, and so on if there are more. So a queen/7/3 would beat queen/6/4. The queens tie so the second highest cards are used to break the tie, and a 7 beats a 6. The third card does not matter in this case because the hand was resolved by the second card.

If you want to know why queen/6/4 is the borderline hand it is because if you raise on queen/6/3 you can expect to lose 1.00255 units, more than the 1 unit by folding. However if you raise on queen/6/4 the expected loss is .993378, less than the 1 unit by folding.
I have been asked several times about the strategy of raising on any queen or better, in other words mimicking the dealer. This is not a bad strategy but you will lose more with it than the optimal strategy above. The house edge playing the mimic the dealer strategy is 3.45%. Raising on everything, or playing blind, results in a house edge of 7.65%.

Player Wins Ties

I have been asked a few times about the probability of tying the dealer in Three Card Poker. This is a valid question because some casinos let ties go to the player. The effect of this rule lowers the house edge in the “full pay” bonus table 1 above from 3.37% to 3.24%. I saw this happen once at the Luxor and the player was paid.

There is a small strategy change to make if the ties go to the player. Under this rule the player should still raise on Q/6/4 or better, plus raise on Q/6/3, only when all three suits are different (this lowers the probability of a dealer flush).

If the player follows the proper raising strategy under the ties win rules then the probability of a tie is 268272/407170400 = 1 in 1517.75 hands. For academic purposes only, if the player always raises then the probability of a tie is 450528/407170400 = 1 in 903.76.

Better known as “The Wizard of Odds”, Michael Shackleford uses math and computer analysis to determine optimal playing strategy for all casino games. His work can be found on his website, WizardofOdds.com with player strategies and probabilities on most of the casino games. Michael lives in Las Vegas and in his spare time likes to college license plates and gamble.You can read more about Michael’s work at his website, WizardofOdds.com

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Three Card Poker

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