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Three Card Poker - FAQ
I have no doubt that Q/6/4 is the optimal strategy for three card poker. Stanley Ko independently came up with the same advice. This strategy is based on a computer program which analyzes all 22,100 possible combinations of the player’s three cards and for each one the remaining 18,424 possible combinations of the dealer's three cards.
If you follow the dealer's strategy, then you will be folding on some hands which have an expected return of more than -1 (the return by folding). Overall, however, it shoudn't make a big difference.
I'll just work out the king high first and briefly show the formula for the other two. The probability will be the number of king high hands divided by the total number of hands. The number of ranks less than king is 11. The king high hand must have two different of these ranks. The number of ways to arrange 2 out of 11 is combin(11,2) = 55. However, one of these combinations is king-queen-jack, resulting in a straight, so subtracting that combination there are 54 left that do not form a straight. Next, there are four suits for each rank, or 43=64 possible combinations of suits. However, four of these 64 result in a flush so there are 64-4=60 combinations of suits left. So, the total number of king high combinations is 54*60=3240. There are a total of combin(52,3)=22,100 to arrange 3 cards out of 52. So, the probability of forming a king high is 3,240/22,100 = 0.1466063 . The probability for ace high is: (combin(12,2)-2)*(43-4)/combin(52,3)=0.1737557. Note the -2 instead of -1 because of both the a-2-3 and q-k-a straights.
The probability for queen high is: (combin(10,2)-1)*(43-4)/combin(52,3)=0.119457.
T.T. from Clarkston, Michigan
You're right, the house edge comes from the player having to act first. If both the player and dealer fold, the player loses.
G.M. from Monmouth County, New Jersey
Please see my hole carding strategy for Three Card Poker. Following the strategy, you will enjoy a 3.48% advantage!
John from Crestwood, Illinois
In the long run it doesn’t matter what you do. As I have said numerous times measured by long term results all betting systems are equally worthless. Systems in which you chase loses with bigger bets increase the probability of a modest short term win but at the cost of even bigger losses when your luck is at it’s worst.
Bruce from Williston, Vermont
The probability of getting three queens in one hand is combin(4,3)/combin(52,3) = 0.000181. The probability of doing it twice in a row is 0.0001812, or 1 in 30525625. The probability of them being the same three suits both times is 0.0001812/4, or 1 in 122102500.
Ralph from Harpster
The probability of getting a straight flush on the first hand is 4*12/combin(52,3) = 48/22100 =~ 0.0022. The probability that the next hand will be exactly the same is 1/22100. So the answer is (48/22100)*(1/22100) = 48/488410000, on 1 in 10,175,208. This is a 1.37 more likely than hitting a 6/49 lottery, which has a probability of 1 in 13983816.
Eddie from West Memphis, Arkansas
Good question. In full play Three Card Poker the house edge on Pairplus is 2.32% and on Ante & Play is 3.37%. However the element of risk on Pairplus is still 2.32% while in Ante & Play it is 2.01%. I believe if comparing one game to another the element of risk is more appropriate. In other words comparing the expected loss to the total amount bet. In this case Ante & Play has the lower element of risk and is thus the better bet. So I would disagree with the writer of the article you mention. According to my house edge index the element of risk in Let it Ride is 2.85%, higher than that of Ante & Play.
Ruby from Tacoma, USA
Good question. In Stanley Ko’s booklet Mastering Three Card Poker he says that if you had a concealed computer to take maximum advantage of the information then seeing the first hand would lower the house edge from 3.37% to 3.31% on the second hand. Even if you could see all seven hands at the table the house edge would still be 2.32%.
Mark from Jacksonville, Florida
I asked this very question on one of my homework assignments for my casino math class at UNLV. Although the house edge is generally higher on the Ante & Play it is also the better bet. The reason is that it has a lower element of risk, in other words the ratio of expected loss to the total amount bet.
Arthur from West Orange, New Jersey
The optimal ratio is to bet 100% on the ante and 0% on the Pairplus. Assuming full pay rules the element of risk is 2.01% on the Ante and 3.37% on Pairplus. Your goal should be to minimize the element or risk as much as possible. Be warned that every other player will bet on the Pairplus and will ridicule those who don’t go along. Once I bet $50 on the Ante only and got a straight flush, which would have paid $2000 on the Pairplus. The other players had a good laugh at my expense but I had no regrets.
Paul from Bradford, England
The probability of any hand less than a pair is the product of the number of ways to pick 3 different ranks out of 13, less 12 for the consecutive ranks that result in a straight, and the number of ways to pick a suit 3 different times, less 4 for picking the same suit each time. So the total combinations for ace-high or less is (combin(13,3)-12)*(43-4) = 16,440.
Now let’s look at the combinations for a jack high or less. We have omitted 3 ranks so there are 3 ranks to choose from among 10. However 8 of these combinations result in a straight (2/3/4 to 9/10/J). Again there are 43-4 ways to pick the suits. So the total combinations is (combin(10,3)-8)*( 43-4) = 6,720. The total combinations for Q-A high is simply 16,440-6,720=9,720. For an explanation of the combin function please see my probabilities in poker section.
Joe from Sloatsburg, USA
The probability of a straight is less than a flush with 3 cards. The number of ways to form a flush is 4*(combin(13,3)-12) = 1096. The number of ways to form a straight is 12*(43-4) = 720.
Larry from Silverdale, Washington
There are combin(52,3) = 22100 ways to arrange 3 cards out of 52. So the probability of any given hand matching the last one exactly is 1 in 22100.
The probability is about 1 in 1 in 903.76 but the solution is too involved to explain.
Mark from Jacksonville, Florida
100% Ante and 0% Pairplus. The reason is the Ante bet has a lower element of risk. The house edge is defined as the ratio of the expected loss to the initial wager. The element of risk is defined as the ratio of the expected loss to the average total wager. Although the Pairplus has the lower house edge I believe in comparing one game against another the element of risk should be used. The element of risk on the Pairplus is the same as the house edge, 2.32%, and on the Ante bet the element of risk is 2.01%, assuming full pay rules. Casinos that offer less than full pay take much more from the Pairplus bet, making the Ante even better by comparison. So don’t waste any money on the Pairplus, although the other players will think you’re crazy as a loon.
John from London, England
Thank you. No, the house edge is not affected by the table minimum and maximum. The greater the spread between minimum and maximum bet the greater the volatility but in the long run results will keep getting closer to the house edge. Some people incorrectly believe that setting a maximum bet increases the house edge, but it does not.
Bet 100% on the Ante, because the ante has the lower element of risk.
You’re welcome. To minimize the element of risk you should raise until the point where the expected value of raising is less than the element of risk for the entire game. The expected loss by raising on Q-6-2 is 1.24% and on Q-5-4 is 2.10% to 2.15% (depending on how the cards are suited). So to minimize the element of risk you should raise on Q-6-2 or higher.
Three of a Kind 30:1
Straight Flush 50:1
Thanks. This pay table has a house edge of 5.10%.
The house edge of that pay table is 2.70%.
I have thought about that many times but always shoot down the idea because it would be very high maintenance to keep it up to date. I do know they have full pay 1/4/6/30/40 pay table on the Pairplus in Three Card Poker at the Pioneer in Laughlin. At least they did when I was last there a few months ago. As far as I know every casino in Vegas follows the stingier 1/3/6/30/40 pay table.
If the dealer exposes one of his cards, which happens frequently, the player has a 3.48% advantage if the information is used properly. I explain the proper strategy in my book Gambling 102. However, to answer your question, you would fold Q86 against a queen.
WM from Ventura
Tournament strategy is not my strongest area. However if a flush pays 3, which is usually the case, I would avoid the PairPlus unless you need a miracle in the last few hands. Otherwise the higher house edge and variance will probably grind you down. Stick to the ante bet when you need to make a move.
Cherrice from North Carolina
I strongly believe the makers of the shuffling machines at least attempt to make the shufflers as fair and random as possible. A deliberately gaffed machine I’m sure would violate Nevada law. It is fairly easy to see good x-card hands in x+1 cards. For example the probability of a three of a kind in three cards is 0.235%, and in four cards 0.922%, or almost four times higher.
Linda from Atlantic City
I forwarded this story to Brian, who is a former gaming regulator and current operator. Here is what he wrote.
All of the table limit signs usually have the caveat "management decision is final" - not much comfort to the player, but they’ll fall back on this for justification. In the scenario described, I would have allowed the hand to continue especially if all of the cards were already out. If I had concerns, I would change the deck out after the hand. Many casinos won’t allow 3CP players to even look at their hands until all cards are dealt. This was cutting into my hands per hour so I changed the procedures. Since the potential appeasement payout for a person that receives a good hand and then the shuffler dies is relatively small, I’m willing to take the risk. In Caribbean Stud, no one touches the cards until they are all on the table.
Tim from Deadwood
If you can trust the player to make the raise bet when he raises his own bet then you should put the tip on the ante. However, if the player will never make the raise bet for you, then the tip would carry a 10.11% house edge, making the Pairplus the much better bet.
James from Genting, Malaysia
Yes, you should. If you see the dealer has a 2 to jack the odds favor raising on anything. Using this strategy does result in a player advantage. I get into the details in my book Gambling 102.
Dan from Las Vegas
You’re right. If the player makes an ante bet for the dealer then he doesn’t seem required to match it with a raise bet if he raises his own bet. The optimal strategy on the tip is to raise with K/Q/10 or better. To keep it simple, this is almost the same as ace high. Anything less and the raise portion of the tip is a bad bet. Following this strategy will result in an advantage of 26.09% on the combined tip bets.
Giorgio I. from San Juan
Please see the following table. This table also shows the house edge assumed for player rating purposes. My source is an executive with a major Strip casino here in Vegas, who wishes to remain anonymous.
Hands per Hour and Average House Edge
|Let It Ride||52||2.4%|
|Pai Pow Poker||34||1.96%|
|Single 0 Roulette||35||2.59%|
|3 Way Action||70||2.2%|
"Anonymous" . from Niagara Falls
Don’t worry, I won’t give your name.
Thomas from Austin
Thanks for the kind words. I get asked this question a lot regarding every game where there are two or more bets to choose from. You should bet 100% on the better bet. In the case of Texas Hold ’em bonus the element of risk on the ante is 0.53% and on the bonus bet it is 8.54%, assuming Vegas rules both ways. For comparing one bet to another I believe the element of risk should be used. So in this case, the ratio of ante bet to bonus bet should be infinity, because the bonus bet should be zero. Same thing with Three Card Poker, which is usually the venue this question is asked about. In that game you should bet 100% on the ante and 0% on the Pairplus.
Chris from Coon Rapids
Much as in blackjack, the dealer has the advantage because he gets to act last. If both the player and dealer in blackjack bust, the player loses. In Three Card Poker, if both the player and dealer have lousy hands, the player will fold first, and lose.
Scott from San Diego
Ideally, you should challenge the hand before it is over, while it is still easy to run back the cards. It doesn’t hurt to ask later than that, but you are not entitled to anything. This is getting outside my area of expertise, but the decision whether or not to review the tape would likely depend on the amount of money involved and your value as a player.
p.s. I only play this game because of your book and when one of my sloppy blackjack dealers is dealing Three Card Poker. First time I played I got a straight flush, and the reaction from the table "experts" tearing into me for not playing pair plus was worth the 40 to 1 bet I didn't win. Ignorance is bliss I suppose.
Brock W. from Bible Hill, N.S.
Thank you for buying my book. I heard somebody out there bought a copy. The difference in house edge between the usual 1/4/5 and the 1/3/4 pay tables is 0.46%. So that would lower the player advantage from 3.48% to 3.02% if you can tell the dealer's rank exactly, 1.95% if you can tell only ace/paint/no paint, and increase the house edge to 2.89% if you can tell paint/no paint only.
I've suffered in silence the Pairplus lecture many times, so I know how you feel.
For more information, please see my page on flashing Three Card Poker dealers.
Joshua K. from Oceanside
That rule change is worth 2.49% in the player’s favor, lowering the house edge from 4.30% to 1.80%. The subraction is not exactly 2.5% less, due to rounding.
Richard from Bremerton, WA
I assume if the dealer doesn’t qualify, then the player wins $10. In this case, the player should always raise. According to my calculations, the value of this coupon is $2.57.
I found these crappy roulette odds from the ferry between England and the Netherlands:
- 1 Number: 30 to 1
- 2 Numbers: 15 to 1
- 3 Numbers: 10 to 1
- 4 Numbers: 7 to 1
- 5 Numbers: 5 to 1
- 6 Numbers: 4 to 1
And it’s American-style (double-zero), despite the fact that the ferry is going between two European countries. What are the odds? — Spanky McBluejay
Shame on that ferry operator. The house edge varies from 13.16% to 21.05%, as follows.
Netherlands/England Ferry Roulette
Jesse from Scottsdale
I’ve never seen that before; thanks for the information. The house edge on the ante bet, with those ante bonuses, is 4.75%.
Indeed, in my experience dealers never pay the Ante bonus, as they are supposed to, when the dealer wins. I’ve seen this happen several times, and every time I had to summon the floor supervisor to get paid. To answer your question, the 2009 Gaming Revenue Report says that Nevada casinos earned $134,181,000 from Thee Card Poker in 2009. The house edge in Three Card Poker is 3.37% on the Ante and 7.28% on the Pairplus.
Let's assume the player bets both equally, for an average house edge of 5.325%. Dividing the profit by the house edge gives us the handle (total amount bet) of $2,519,830,986. Again, let's assume half of that, or $1,259,915,493, was bet on the Ante.
I roughly estimate the Ante bonus error costs the player 0.00072 of his Ante bet, on average, assuming the dealer always makes that error. So, over $1.259 billion bet, the cost of that error would be about $909,000 per year. However, to be fair, I'll say 25% of time the dealer won’t make that mistake, lowering that figure to about $682,000 per year. While that is a small fraction of the total amount bet in Three Card Poker, it is still not an insignificant amount of money. Hopefully this will educate players about this frequent error. Don't be afraid to throw the challenge flag if it happens to you or another player.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.
Yes! Let’s consider the following situation:
Dealer hits soft 17
Player has A,6
Dealer shows 2
According to my blackjack appendix 9, the following is the expected value of each play:
So, hitting is the play that results in losing the least amount of money on average for that hand. If the player were to double, the expected value of that error would be -0.004882 - (-0.000274) = -0.004608. According to my blackjack house edge calculator, the house edge under those rules — assuming surrender, double after a split and re-splitting aces — is 0.48%. Usually, some of those options won’t be allowed, increasing the house edge. So, as long as the dealer hits a soft 17 in a 6-deck game, the cost of doubling soft 17 against a 2 is less than the cost of betting the same amount on an additional hand.
You could make your same point in any game that involves raising. For example in Three Card Poker, if you want to minimize the expected loss per hand, then the optimal strategy is to raise on Q64 or better, as I state on my Three Card Poker page. However, if your goal is to minimize the expected loss per total amount bet, then the optimal strategy is to raise on Q62 or better.
This begs the question of why do gambling writers like me base strategy on minimizing the expected loss per original bet, rather than the total amount bet? My answer is that it is mainly out of tradition. That is how the blackjack basic strategy was created, and everybody has kept that methodology out of habit and simplicity. If the recreational player’s goal is to minimize losses over a defined period of time, then he should go with conventional strategies that minimize the expected loss per hand. If the player’s goal is to minimize losses over $x in total bets, then he should make the kind of marginally bad doubles and raises mentioned. I tend to think most players have a time-based goal, favoring the conventional strategies.
This question was raised and discussed in the forum of my companion site Wizard of Vegas.