Three Card Poker

Last update: July 13, 2008

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Rules

1. Player makes an Ante and/or Pairplus bet
2. The dealer gives each player three cards and himself three cards. The player may examine his own cards. The dealer's cards are dealt face down.
3. If the player made the Ante bet, then he has must either fold or raise.
4. If the player folds, then he forfeits his Ante wager.
5. If the player raises, then he must make an additional Play bet, equal exactly to his Ante bet.
6. The dealer will turn over his cards.
7. The dealer needs a queen high or better to qualify.
8. If the dealer does not qualify then the player will win even money on the Ante bet and the Play bet will push.
9. If the dealer qualifies, then the player's hand will be compared to the dealer's hand, the higher hand wins. The order of poker hands is indicated below.
10. If the player has the higher poker hand then the Ante and Play will both pay even money.
11. If the dealer has the higher poker hand then the Ante and Play will both lose.
12. If the player and dealer tie then the Ante and Play bets will push.
13. If the player made the Ante bet and has a straight or higher then the player will receive an Ante Bonus, regardless of the value of the dealer's hand.
14. The Pairplus bet will pay entirely based on the poker value of the player's hand, as shown in the Pairplus section below.

Ante and Play Analysis

Various pay tables are known to exist for the Ante Bonuses. The table below shows the various pay tables known, the contribution to the return for each and, and the overall house edge and element of risk for each pay table.

Mini Royal rules: On March 27, 2008, I saw an Ante Bonus pay table at the Wynn that paid for 0 for a straight, 3 for a three of a kind, 4 for a straight flush, 10 for a Mini Royal (suited AKQ), and 50 for a Mini Royal in spades, plus a tournament entry. Ignorring the value of the tournament, the house edge under that pay table is 6.79%. I also have an unconfirmed report that in some casinos in Arizona the ante bonus is 0 on a straight, 6 on a three of a kind, 8 on a straight flush, and 50 on a mini royal. The house edge on the ante bet under this pay table is 4.75%.

The next table shows the probability and return for each net win under the table 1 ante bonuses.

Overall on the ante the probability of a win is 44.91%, push 0.06%, and loss 55.03%.

Pairplus

The following table shows the probabity and return for the most common pay table of the Pairplus bet. The lower right cell shows a house edge of 7.28%.

The following two tables show various paytables for Pairplus. It used to be that the only table was pay table 1. Now pay table 6 is the norm. The only casino in southern Nevada that I know of to still offer table 1 is the Pioneer in Laughlin. Bodog offers paytable 1 in its online casino.

In 2007 some casinos added a "Mini Royal" to the pay table. A Mini Royal is a suited ace/king/queen.

The following two pay tables show the house edge of various Pairplus pay tables, both with and without the Mini Royal.

Pairplus Pay Tables — With Mini Royal
Hand	Probability	Table 8	Table 9	Table 10
Mini Royal	0.02%	50	200	100
Straight flush	0.2%	40	40	50
Three of a kind	0.24%	30	30	40
Straight	3.26%	6	6	6
Flush	4.96%	3	3	3
Pair	16.94%	1	1	1
Nothing	74.39%	-1	-1	-1
House edge		7.1%	4.38%	1.85%

I have an unconfirmed report that Table 10 is offered at the Borgota in Atlantic City.

Combined

The next table shows the probability and return for net win assuming the player bets equally on the Ante and Pairplus, assuming the most common pay tables of 1/4/5 on the Ante Bonus and 1/3/6/30/40 on the Pairplus. The lower left cell shows an expected loss of -.10649 units on a two unit initial bet. So the house edge of the combined bet would be -.10649/2 = -0.053245, or a house edge of 5.32%

Overall betting equally on the Ante and Pairplus on the pay tables stated above the probability of a win is 32.06%, push 13.23%, and a loss 54.71%.

Strategy

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 mimicing 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

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.

California Rules

In the poker casinos of Los Angeles county Three Card Poker has a few rule changes. For more information see my section on California Three Card Poker.

Macau Rules

The Sands did has a game titled "3 Card Poker." However this game is a simple showdown between the player and dealer, higher hand wins. A player win paid 19 to 20, or even money less a 5% commission. Below is a return table for this game. The lower right cell shows a house edge of 2.50%.

There was also a game in Macau called Q Poker, which is very similar to Three Card Poker.

Tipping Strategy

At many casinos if the player makes a tip for the dealer then the player has the option to call on the tip portion of the bet. For example if the player bets $5 for himself and $1 for the dealer the player may raise his own wager but not the tip. The correct strategy based on the tip alone is to call any hand of king high or less. So on hands of Q/6/4 to K/Q/10 the player should raise his own bet but not the tip. However with less than Q/6/4 there is a conflict of interest. To maximize the total expected value of the player's bet and the tip the player should raise on hands just under Q/6/4. The greater the ratio of tip to bet the more hands under Q/6/4 the player should raise on.

A fantastic advantage play, in collusion with the dealers, would be to bet the table minimum on the bet and the table maximum on the tip. Then raise on ace high or better, otherwise call. This would result in a player advantage on the tip of 26.09%.

Appendix

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Links

German translation of this article.