Three Card Poker

Last update: July 6, 2010

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Rules

Player makes an Ante and/or Pairplus bet
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.
If the player made the Ante bet, then he has must either fold or raise.
If the player folds, then he forfeits his Ante wager.
If the player raises, then he must make an additional Play bet, equal exactly to his Ante bet.
The dealer will turn over his cards.
The dealer needs a queen high or better to qualify.
If the dealer does not qualify then the player will win even money on the Ante bet and the Play bet will push.
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.
If the player has the higher poker hand then the Ante and Play will both pay even money.
If the dealer has the higher poker hand then the Ante and Play will both lose.
If the player and dealer tie then the Ante and Play bets will push.
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.
The Pairplus bet will pay entirely based on the poker value of the player's hand, as shown in the Pairplus section below.

Hand Order and Probabilities in Three Card Poker
Hand	Combinations	Probability
Straight flush	48	0.002172
Three of a kind	52	0.002353
Straight	720	0.032579
Flush	1096	0.049593
Pair	3744	0.169412
Ace High or Less	16440	0.743891

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. The vast majority of tables follow pay table #1.

Ante Bet Bonus
Hand	Probability	Table 1	Table 2	Table 3	Table 4
Straight flush	0.002172	5 to 1	4 to 1	3 to 1	5 to 1
Three of a kind	0.002353	4 to 1	3 to 1	2 to 1	3 to 1
Straight	0.032579	1 to 1	1 to 1	1 to 1	1 to 1
House Edge		3.37%	3.83%	4.28%	3.61%
Element of Risk		2.01%	2.28%	2.56%	2.16%

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

Ante Return Table
Win	Combinations	Probability	Return
7	617044	0.001515	0.010608
6	931972	0.002289	0.013733
5	289104	0.000710	0.003550
3	8976452	0.022046	0.066138
2	91100696	0.223741	0.447482
1	80955780	0.198825	0.198825
0	249216	0.000612	0.000000
-1	132923304	0.326456	-0.326456
-2	91126832	0.223805	-0.447610
Total	407170400	1.000000	-0.033730

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

Mini Royal rules

Some casinos now offer a line item for a Mini Royal in the Ante Bonus pay table. A Mini Royal is a suited AKQ. Following are pay tables I have seen or heard to exist. The bottom row shows the overall house edge on the Ante bet for the given pay table.

Mini Royal Summary Table
Hand	Table 1	Table 2	Table 3
Mini Royal in Spades	50	50	10
Mini Royal	10	50	10
Straight flush	4	8	5
Three of a kind	3	6	4
Straight	0	0	1
House Edge	6.79%	4.75%	3.28%
Element of Risk	4.06%	2.84%	1.96%

Mini Royal Ante Bonus locations:

Pay Table 1: Wynn, Las Vegas
Pay Table 2: Casino Arizona
Pay Table 3: Borgata, Atlantic City; Galewind Internet casinos

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%.

Pairplus Return Table
Hand	Pays	Combinations	Probability	Return
Straight Flush	40	48	0.002172	0.086878
Three of a Kind	30	52	0.002353	0.070588
Straight	6	720	0.032579	0.195475
Flush	3	1096	0.049593	0.148778
Pair	1	3744	0.169412	0.169412
All Other	-1	16440	0.743891	-0.743891
Total		22100	1	-0.07276

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. The Borgota in Atlantic City offers pay table 11 as the "Three Card Bonus Bet" in Let it Ride.

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 — Without Mini Royal
Hand	Probability	Table 1	Table 2	Table 3	Table 4	Table 5	Table 6	Table 7
Straight flush	0.22%	40	35	40	35	50	40	40
Three of a kind	0.24%	30	33	25	25	30	30	30
Straight	3.26%	6	6	6	6	6	5	6
Flush	4.96%	4	4	4	4	3	4	3
Pair	16.94%	1	1	1	1	1	1	1
Nothing	74.39%	-1	-1	-1	-1	-1	-1	-1
House edge		2.32%	2.70%	3.49%	4.58%	5.10%	5.57%	7.28%

Pairplus Pay Tables — With Mini Royal

Hand Probability Table 8 Table 9 Table 10 Table 11

Mini Royal 0.02% 50 200 100 50

Straight flush 0.2% 40 40 50 40

Three of a kind 0.24% 30 30 30 30

Straight 3.26% 6 6 6 6

Flush 4.96% 3 3 3 4

Pair 16.94% 1 1 1 1

Nothing 74.39% -1 -1 -1 -1

House edge 7.1% 4.38% 4.2% 2.14%

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%

Combined Return Table
Win	Combinations	Probability	Return
-3	77992716	0.191548	-0.574644
-2	132652800	0.325792	-0.651584
-1	12108216	0.029737	-0.029737
0	53876136	0.132318	0.000000
1	39406040	0.096780	0.09678
2	20926080	0.051394	0.102788
3	36171912	0.088837	0.266512
4	6135276	0.015068	0.060272
5	13064048	0.032085	0.160425
7	18288	0.000045	0.000314
8	4001004	0.009826	0.078611
9	8975484	0.022044	0.198392
32	3312	0.000008	0.000260
35	288960	0.000710	0.024839
36	665776	0.001635	0.058865
43	968	0.000002	0.000102
45	144	0.000000	0.000016
46	266196	0.000654	0.030073
47	617044	0.001515	0.071226
Total	407170400	1	-0.106490

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%.

Three Card Poker Progressive

Three Card Poker Progressive is a side bet I noticed at the Fitzgeralds in Las Vegas on February 4, 2009. It is a $1 wager that pays according to the pay table below. All wins are on a for one basis, in other words the player does not get back his original wager, even if he wins. The envy bonus pays if the player made the bet, but another player gets a mini royal.

Three Card Poker Progressive — Pay Table
Hand	Pays	Envy Bonus
Mini Royal in spades	Jackpot	$100
Mini Royal	$500	$25
Straight Flush	$100	$0
Three of a Kind	$90	$0

The return depends on the jackpot amount and number of other players. To be specific, it is 47.87%, plus 4.52% for each $1000 in the jackpot, and 0.79% per number of other players. To breakeven the meter needs to be $11,520, less $175 for each other player.

Prime

The "Prime" is a side bet I noticed at the Wynn on July 2, 2010. If the player's three cards are all the same color it pays 3 to 1. If all six player and dealer cards are the same color, then it pays 4 to 1. The following return table shows a house edge of 3.62%, which for a side bet isn't bad.

Prime
Hand	Pays	Combinations	Probability	Return
6 Same Color	4	9,209,200	0.022618	0.090470
3 Same Color	3	86,595,600	0.212677	0.638030
Loser	-1	311,365,600	0.764706	-0.764706
Total		407,170,400	1.000000	-0.036206

Red Hawk Bonus

I have an unconfirmed report that the Red Hawk casino, near Sacramento, offers the following $1 side bet. All wins are on a for one basis, meaning the players never get the original bet back, even if they win. The player can win based on his own hand, plus envy bonus wins based on the hands of other players. The following return table shows the pay table for players own cards. The lower right cell shows a return of 57.47%.

Red Hawk Bonus — Player's Hand
Hand	Pays	Combinations	Probability	Return
Mini Royal in spades	4000	1	0.000045	0.180995
Mini Royal	1000	3	0.000136	0.135747
Straight Flush	100	44	0.001991	0.199095
Three of a Kind	25	52	0.002353	0.058824
Loser	0	22000	0.995475	0
Total		22100	1	0.574661

The next table shows the value of the Envy Bonus per additional player. The lower right cell shows that the return increases by 0.79% per additional player at the table.

Red Hawk Bonus — Envy Bonus
Hand	Pays	Combinations	Probability	Return
Mini Royal in spades	100	1	0.000045	0.004525
Mini Royal	25	3	0.000136	0.003394
Straight Flush	0	44	0.001991	0.000000
Three of a Kind	0	52	0.002353	0.000000
Loser	0	22000	0.995475	0.000000
Total		22100	1	0.007919

The next table shows the house edge according to the total number of players.

Red Hawk Bonus — House Edge
Players	House Edge
6	38.57%
5	39.37%
4	40.16%
3	40.95%
2	41.74%
1	42.53%

6 Card Bonus

The "6 Card Bonus" is a side bet in Three Card Poker I noticed at Harrah's in Las Vegas on September 18, 2009. It pays based on the highest five-card poker hand that can be composed out of the player's and dealer's three cards. The following table shows the probability and return from each hand. The lower right cell shows a house edge of 15.28%.

6 Card Bonus
Hand	Pays	Combinations	Probability	Return
Royal flush	1000	188	0.000009	0.009234
Straight flush	200	1656	0.000081	0.016268
Four of a kind	50	14664	0.000720	0.036014
Full house	25	165984	0.008153	0.203826
Flush	15	205792	0.010108	0.151626
Straight	10	361620	0.017763	0.177626
Three of a kind	5	732160	0.035963	0.179817
Loser	-1	18876456	0.927202	-0.927202
Total		20358520	1	-0.152790

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 Ante but win 5.29% on the Raise.

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

I have been asked a few times about the probability of tieing 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%.

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.

Ultimate Three Card Poker

The Silver Legacy in Reno has a variation, in which the player may raise up to 3X his bet. For all the details, visit my page on Ultimate Three Card Poker.

Macau Rules

In 2007 the Sands had 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%.

3 Card Poker - Sands Macau Rules
Event	Pays	Probability	Return
Win	0.95	0.499434	0.474462
Lose	-1	0.499434	-0.499434
Tie	0	0.001132	0
total		1.000000	-0.024972

In 2007 there was also a game in Macau called Q Poker, which is very similar to Three Card Poker.

In August, 2009, many casinos had traditional Three Card Poker, only they call it Fortune Three Card Poker in Macau. For more information, please see the page on Fortune Three Card Poker at WizardOfMacau.com.

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

To see how to calculate the probabilities in Three Card Poker yourself visit my Three Card Poker Appendix.

Play for Fun

Practice playing Three Card Poker for free on my Java applet.