Pai Gow Poker - FAQ

Sometimes in pai gow poker the dealer sets aside a dragon which is offered to each player in turn. This is like playing two hands and is likely what you are observing. However, when playing the dragon hand the rules generally specify that it must be set according to the house way.

Even if the player could use information from 14 cards when setting his hand, I doubt it would help that much, and making proper use of the information would be complicated.

No, you should not make the Fortune side bet, or any other side bet. Generally speaking, they are sucker bets that carry a high house edge. For more information on the Fortune side bet visit my pai gow poker side bets section.

Generally, no, the odds do not change depending on the number of players. The only exception I can think of is side bets like Fortune in Pai Gow Poker, which feature an envy bonus, and thus pay better according to the number of players.

Not counting a three of a kind and two pairs, the following are the ways to get a three pair and number of combinations.

No wild card: combin(13,3)*10*6³*4 =2471040
Wild card used to compete pair of aces: combin(12,2)*10*6²*4² = 380,160
Wild card used as singleton ace: combin(12,3)*6³ = 47,520

The total number of combinations is 2,898,720. This is less than half of the 747,0676 combinations for a three of a kind.

Gambling is supposed to be fun, so if you think you get 39.5 cents worth of fun every $5 you bet, then you should play it. That is how much you will give up in the house edge, assuming no other players.

There are two ways to arrange the ranks for form a 9 high hand, the one you mentioned and 2-3-4-6-7-8-9. The number of suit combinations without forming a flush is 4⁷-4*(combin(7,5)*3^2+6*3+1) = 15,552. So the probability of a 9 high hand is 2*15,552/combin(53,7) = 31,104/154,143,080, or 1 in 9,911. If you were to play just five times the probability of getting 2 9-high hands would be 1 in 9,826,685. That this happened is a coincidence I believe as opposed to a fault in the random number generator or the coding of the program.

I have always wondered this myself. It is probably just one of those weird things like why we spell colonel the way we do.

Sometimes the two-card hand is so bad that it is better to put the third highest card as a kicker in the five-card hand. There is no rule of thumb to advise exactly when to do so but I have noticed it tends to happen when the second highest card is rather low. The logic my program uses is a large array of probabilities that each hand will win and goes with the hand with the greatest possible sum of probabilities that won't spoil the hand.

Thanks for your kind words. Actually, I have been asked about teaching a coarse on the mathematics of gambling at UNLV. pai gow poker is not my game, so I don't follow the details very closely. I do know, as you stated, that some rotate and some zig-zag the banker between the players and dealer. However I don't keep track of who does it which way, sorry. I've also seen that progressive side bet at lots of casinos around town. Again I don't keep track of who specifically has it. However that is a great idea of banking against it, I have never thought of that. Sorry I wasn't of much help.

Good question. I assume player banking is still allowed. The 5% commission at the Nevada casinos costs the player 1.46%. The house edge in your game depends on how much you bet. It is lowest for a $100 bet, at 1%. I’m ignoring the rule about everybody pushing, that will lower the edge even more. Much like the card rooms of in the Los Angeles area I think there is potential to beat the house edge if you bank enough of a percentage of the action.

There are 32 possible natural straight flushes (4 ranks times 8 possible spans of 7 cards). There are combin(53,7) = 154143080 possible ways 7 cards can be drawn out of 53. So the answer is 32/154143080, or 1 in 4816971.

Yes, I have done a similar analysis. It can be found in my pai gow poker appendix 2. There are some important difference between my strategy and his. Wong’s is much more detailed, considering the highest two singletons, while mine considers only the single highest. Wong also differentiates between whether the player is banking or not. Finally Wong’s book is based on the California game with no 5% commission, unlike my strategy. I do trust Wong’s work and don’t disagree with his table.

The probability of two specific players ties with a royal flush and any two other cards are about 1 in 290 billion.

Pai gow poker is the least volatile and on average keno is the most.

A 4% commission lowers the house edge by 0.29%.

They probably don’t have to. Once at the Tropicana in Atlantic City their pai gow poker rules said the house way was available upon request. So I requested it and they ran out of public copies and couldn’t show me a house copy because it didn’t have the Gambler’s Anonymous disclaimer on it. In my opinion the player should always have the right to know the rules of a game, but unfortunately all gaming authorities seem to think differently.

The various house ways are all very similar and only differ in rare or borderline plays. I have often heard dealer’s comment that their casino uses a conservative house way that tries to balance the hands, resulting in more pushes. However I would doubt if anyone has ever done a comparison study.

This would make me furious too. While not banking it should not make any difference who is banking. I have never heard of an etiquette rule written about this situation but it falls under a breach of common courtesy in my opinion.

The player loses 0.42% because of the lower royal flush pay-out. However I show the probability of a three pair to be 1.88%. So overall these changes decrease the house edge by 1.46%.

Thanks. I just asked a dealer and he confirmed that the house would win that hand because the joker would be used as a king. The general rule is the joker can substitute for any specific card not already in the same hand as long as it completes a straight, flush, or straight flush. Otherwise it is treated as a fifth-suit ace, thus allowing for the possibility of five aces.

Thanks. Considering both odds of winning and bankroll preservation I think the choice comes down to blackjack (which favors winning) and pai gow poker (which favors bankroll preservation). If you can find a $5 table in either game I would go with blackjack. If the minimums are $10 then I would go with pai gow poker.

I think this would put a small dent in the house edge, assuming you know how to make the correct strategy adjustments, but will not come anywhere close to overcoming it.

The rule of thumb when it comes to comps is that the casinos give back some percentage, usually one-third. So if your goal is to get the room with as little expected loss as possible then whatever game offers the lowest house edge is what you should play. You will probably earn that room faster and with less bankroll volatility playing pai gow or pai gow poker. However the house edge is higher so your expected loss will be greater than in blackjack. In my opinion you should play whatever you would play if there were no comps at all. Then consider comps as icing on the cake.

The casino had the right to do this. However in my opinion it was a bad business decision. Not only did the casino waste time resolving this mess but as you point out it resulted in bad feelings on the part of all players. This just goes to show the folly of following rules religiously. Personally I think rules should be weighed against common sense.

The number of combinations for a three pair without the joker is combin(13,3)*10*combin(4,2)^3*4/combin(52,7) = 2,471,040. The number of combinations of a three pair with the joker being used to complete a pair of aces is 23776. The number of combinations of a three pair with the joker as the singleton is 61776. So the total combinations are 2556592. Out of total combin(53,7)=154,143,080 possible the probability of a three pair is 1.659%. So changing a three pair from a loss to a win of 1 to 1 decreases the house edge by 3.32%. Assuming the standard pay table on the other hands this would sway the odds in the player's favor with 3 or more other players.

My two pair rule is optimized to play against the house way. However I think it is probably just any reasonable strategy. For example I would use it when banking against other players. The reason the casinos use a more complicated and less powerful rule is probably out of tradition. Whoever invented the game probably came up with that strategy rather arbitrarily and it since become a hard habit to break. Two other rules I find ridiculous are counting A2345 (known as "the wheel") as the second highest straight and bothering to state an exception in the house way that if the dealer has five aces with a pair of kings to play the pair of kings in the low hand. The probability of getting this hand is 1 in 25,690,513. In my estimation this hand may have come up about 100 times in the history of the game, but has probably never affected the outcome of a hand compared to the alternative of playing a full house in the high hand. Yet every single dealer to have dealt the game had to be bothered with learning the exception.

From my pai gow poker section we have the following probabilities.

Player wins both 28.61%
Tie 41.48%
Banker wins both 29.91%

So you are getting away with a 4% commission. As player your expected value is .2861*0.96 - .2991 = -2.44%. As banker (going one on one) is .2991*0.96 - .2861 = 0.001036. So a 4% commission lowers the house edge by about 0.3%.

P.S. In May, 2008, a reader wrote to say that the Borgata does use quarters in pai gow poker, and forces bets to be in increments of $5.

The odds are the same. However it will be less volatile for both of you to play at half the bet size.

You're welcome. Here is the number of ways to make a three pair:

No joker: combin(13,3)*10*combin(4,2)³*4 = 2,471,040
Joker used in a pair of aces: combin(12,2)*10*4²*combin(4,2)² = 380,160
Joker used as singleton: combin(13,3)*combin(4,2)³ = 61,776

The total number of possible combinations is combin(53,7) = 154,143,080. So the probability of a three pair is (2,471,040+380,160+61,776)/154,143,080 = 0.0189. So, changing a three pair from a loss to a 1 to 1 push would reduce the house edge by 1.89% .

According to my sources in Washington state many casinos waive the 5% commission if the player makes the Fortune side bet. The most common pay table for the Fortune bet in Washington I’m told is 2/3/4/5/25/50/150/400/1000/2000/8000, which has a house edge of 7.83% less 1.07% for each additional player at the table. So the expected loss of a $5 Fortune bet is 39.14 cents less 5.34 cents for each additional player. The following table shows the breakeven point between making and not make the Fortune bet according to the number of players at the table, including yourself.

• 1 player: $27.36
• 2 players: $23.63
• 3 players: $19.90
• 4 players: $16.17
• 5 players: $12.44
• 6 players: $8.71
• 6 players: $4.98

For example, if there are four players (including yourself) you should make the Fortune bet if your pai gow poker bet is $17 or more, and not if it is $16 or less.

Thanks for the kind words. I always like to hear from real actuaries out there. My opinion is that it is very bad etiquette to pull back a bet when another player is banking unless the player banking doesn’t mind. Sometimes there is collusion between players to not bank against each other, to reduce losses and volatility. Absent that, refusing to play against a player banker denies the banker the favorable odds the dealer usually enjoys. That just isn’t cool to happily give your money to the casino but not a fellow player. It would be like refusing to give somebody your luggage cart at the airport when you were done with it, even if offered the 25-cent refund.

I never play the dragon hand myself because it always loses on copies. Regardless of what is in your hand you have the same odds of making any given hand as the dealer, assuming the same strategy. So if you must play it you may do so whenever you like.

The break even point is $102,680.24. I just added an analysis of this bet to my section on Pai Gow Poker side bets.

Here in Vegas they always seem to tuck the cards under your bet when you bank and don’t allow you to touch them until everybody else has finished setting their hands. If you try to look at them early the dealer will give you a sharp rebuke. However in your casino I wouldn’t hesitate to try to take a peek if the banking player is setting his hand early. Just try to be discreet. I would liken it to a dealer in blackjack flashing his hole card. That is great information to have and it is perfectly legal to use it.

Yes! The following table shows the house edge as both player and banker according to the number of other players. The last column shows the ratio of overall action as banker to player to break even. So with seven players, or six opponents, you would need to bet 13 times more as banker than player.

Required Banker Ratio to Break Even in Pai Gow Poker

Players	House Edge Banker	House Edge Player	Breakeven Ratio
2	0.2%	2.73%	N/A
3	-0.02%	2.73%	136.5
4	-0.1%	2.73%	27.3
5	-0.15%	2.73%	18.2
6	-0.19%	2.73%	14.4
7	-0.21%	2.73%	13

From my pai gow poker section, the probability of the banker winning is 29.91%. So instead of paying the usual 5% commission as player you are paying 4.76%. That will lower the house edge by 0.2991*(0.05-0.0476) = 0.07%.

In pai gow tiles, it is always obvious which hand is higher. You don’t need to know where to place the high hand. Just make any two stacks of two tiles each, and the dealer will figure it out. I have never once heard of a pai gow tiles tournament. However, the Las Vegas Hilton has an annual pai gow poker tournament. I’m not certain whether you can request the house way, but I would bet against it. It is a very strict rule in most tournaments that the dealer can not give advice of any kind.

For the benefit of other readers, a "pai gow" hand is one with seven singletons, and no straight or flush is possible. The probability that the dealer will have the same seven ranks, without using the joker, is 3⁷/combin(46,7) = 2,187/53,524,680 = 1 in 24,474.

With no commission, the banker has a 1.3% advantage, and all others have a 1.3% disadvantage, assuming the player follows the casino’s house way. If the player banks half the time, then the overall house edge is exactly 0%. If the player banks 1 in 7 hands, then the overall house edge is 0.93%. If the player banks 1 in 14 hands, then the overall house edge is 1.11%. If the ratio of hands that you bank is b, then the overall house edge is 1.2% - 2.4% × b.

Sometimes in Washington, the player will be required to make the Fortune side bet to have no commission. I address this rule in my Ask the Wizard column #159.

To answer the second question first, the banker must have enough chips on the table to cover all bets. If he doesn’t, the dealer will give him the choice to buy more or forfeit his turn to bank.

As for the first question, the table limit still applies when a player is banking. It would seem to be good business to allow any bet, because the casino will stand to get 5% of a larger amount. I asked about this at three different casinos. The following is what I was told, in the order I asked:

Casino 1: The Gaming Control Board needs to approve increases in the maximum bet, which they can not do on short notice.

Casino 2: The Gaming Control Board has nothing to do with it. Instead, a casino vice president needs to authorize any increase in the maximum bet, and it is generally only done for known good customers.

Casino 3: Casinos don’t need Gaming Control Board approval to raise the maximum bet on a table. My source hadn’t heard of a casino allowing unlimited bets in the case of player banking and added, conceptually, there isn’t any exposure for the casino, so there wouldn’t be a reason to preclude it.

I would add that in my many hours of playing pai gow, I have never once seen anything close to this situation come up. Usually, players don’t like to bet against other players, and the maximums are sufficiently high that players rarely bump up against them, regardless of who is banking. However, if the situation happened often enough, I think casinos would indeed re-think their policy and allow unlimited bets.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

The rules of etiquette are not firmly established, so I can only tell you my opinion. First, if you want to bank, then you are perfectly entitled to when it is your turn. Sometimes other players will moan about it, or blatantly complain about it in Chinese, but don’t let them make you feel ashamed to invoke your right to bank. Second, if another player is banking, and you want to bet, then you don’t have to ask anybody’s permission to.

However, if you want to go beyond the call of duty, it would be good to talk to the other player to ensure both of you are happy. If you ask to bank, then you risk a confrontation if the other player says he prefers that you don’t. Usually you can get an idea what he will say anyway. If you think he won’t object, then I would ask. If you wish to bet while another player is banking, it would be a nice thing to do if you didn’t bet more than he was comfortable with, while still betting enough to make it interesting for you. I think it is bad manners to over-bet, in an effort to cause a banking player to back down. This is something I’ve seen happen several times.

To summarize, if the other player you would be betting against looks like a reasonable person, then I would try to work out something mutually agreeable. If he seems like the "to hell with you" type, then I would just do whatever you want.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

My pai gow poker appendix 1 is useful for questions like this. To answer the question, add the low and high hand power ratings for all viable ways to play the hand. The following table shows the sum of the power ratings (for not banking) for both viable ways to play the hand. It shows that breaking up the royal, while painful, is the much better play.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

Based on a simulation of 7.7 billion hands, assuming the player follows the house way, the probability of a tie in the front (low) hand is 2.55%, or 1 in 39.24. The probability of a tie in the back (high) hand is 0.038%, or 1 in 2,637. The probability of a double tie is about 1 in 78,200.

Good question. The house way says to split up five aces, playing two aces in the low hand. The exception is to retain the five of a kind in the high hand if you can play two kings in the low. My pai gow poker page indicates seven house ways from Las Vegas and Atlantic City, and all of them include this exception. The three from outside those two cities do not.

The probability of getting four aces, the joker, and two kings is 1 in 25,690,513. Assuming the dealer is banking, the only situation where this exception would help is if the player had a four of a kind or better in the high hand. The probability of that is about 1 in 300. The probability of both happening is about 1 in 7.6 billion.

According to the Nevada Gaming Control Board, there were 306 pai gow poker tables in Nevada in 2009. If we generously assume 60 hands per hour, two players per table, and every table is open 24/7, then it would take 23.7 years for this exception to occur and make a difference in the outcome.

So the casinos are requiring every single pai gow poker dealer to memorize this exception when it may never have made a difference in the outcome in the history of the game. I speculate the same person who thought of adding that rule was the same person who decided that the A2345 straight, known as "the wheel," is the second highest straight.

The probability of a three of a kind at 4.98% is lower than that of a straight at 7.16%, due to the wild card rules. In Shuffle Master’s Fortune side bet, a Three of a Kind pays more. While they are allowed to have any rules they like for their own games (subject to the approval of regulators), I doubt Shuffle Master would have the audacity to change the rank ranking of the base game. Not only would the change be un-poker like, but it could result in untold complaints by disaffected players. I suspect the supervisor is confused with the three of a kind paying more on a side bet and incorrectly assumes that means it ranks higher in the base game.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

There doesn’t seem to be any unbeatable hand when not banking. So let’s consider two hands. The first can’t be beaten or tied in the high. The second can’t be beaten or tied in the low.

(1) AAAAW/KK (W=wild)

The five aces cannot be beaten in the high, but the pair of kings could be tied in the low. The number of combinations with two kings is combin(44,5) = 1,086,008. The number of total combinations of 7 cards out of the 46 left is 53,524,680. So the probability of two kings is 1,086,008/53,524,680 = 2.03%. With many of those combinations the dealer will not play the KK in the low. According to my Pai Gow Poker Appendix 1, the dealer will have a pair of aces or less in the high 74.93% of the time. That means he will make a two pair or better, allowing him to play the kings in the low, 25.07% of the time. So the probability the player will get two kings, and be able to play them in the low, is 2.03% × 25.07% = 0.51%, or 1 in 196.

(2) AWQJT (suited)/AA

The royal can be tied in the high, but the aces cannot be beaten or tied in the low. Here I’m using the wild to substitute for a king to make a royal, but it could also have substituted for the Q, J, or T. I don’t want it to substitute for an ace, because then there would be two aces left in the deck for the dealer to tie the low. This way, the dealer can only tie the high with another royal. What are the odds of that? There are three suits left for the royal, and the other two cards could be anything. So the number of combinations that push is 3×combin(41,2) = 2,460. I get the 41 from 46 cards left in the deck after the player removed 7 from the original 43, and then subtract 5 more for the 5 cards in the dealer’s royal. We have combin(46,7) = 53,524,680 in the denominator. So the odds of a tie with wild royal/AA is 2,460/53,524,680 = 0.004596%, or 1 in 21,758.

Before some perfectionist writes to me, there may be some bizarre situations where the dealer doesn’t play the hand the way I intended. I’m not looking to get an exact probability of each situation, but to substantiate why I think wild royal/AA is the best hand you can get in pai gow poker when not banking.

This question was raised and discussed in the forum of my companion site Wizard of Vegas.

The following table shows the possible outcomes, assuming the player is not banking, and the Trump Plaza house way. The lower right cell shows a player advantage of 16.09%.

At the risk of making the advantage player community angry again, I'll just say there are some situations where the astute player can make a wager knowing what his first card will be.

The following table shows the probability of each card and the advantage, if greater than zero, when it is the first card. The conditional return is the expected win, given what the indicated first card is. The expected return is the product of the probability and conditional return columns.

The above table shows that if the player plays only when the first card is a queen or higher his advantage per hand seen is 1.81%. The player will make a bet 24.52% of the time. The advantage per bet made is 7.40%.

Note how the advantage with a queen is 0.05% only. If we don't play those hands, then the table looks like this.

The above table shows that if the player plays only when the first card is a king or higher, then his advantage per hand seen is still 1.81%. The player will make a bet 16.98% of the time. The advantage per bet made is 10.66%.

Here is that table if the player plays aces or the joker only.

This shows that if the player plays only when the first card is an ace or joker, then his advantage per hand seen is still 1.52%. The player will make a bet 9.43% of the time. The advantage per bet made is 16.07%.

See discussion about this question in my forum at Wizard of Vegas.

Pai gow poker was a badly designed game from day one and decades later nobody has bothered to clean it up. Flaws to pai gow poker include:

• An overly confusing and complicated house way.
• The ridiculous rule that the A2345 straight is the second highest.
• Option for player banking and co-banking, which almost nobody, except me, invokes.

When I briefly worked for a major Strip casino, that shall remain nameless, I offered to create a house way that would have been shorter, more powerful, and clearly covered every possible situation. Not just for pai gow poker, but tiles as well. Of course, that suggestion was shot down without comment.

Now that my rant is out of the way, I'll try to answer your question. While there are some differences from one house way to another, they are all more or less organized the same way. The Foxwoods house way is typical. Using your hand as an example, it isn't clear whether to treat is as a straight hand and play it as AKQJT/TT or a full house hand and play it as TTTJK/AA.

Not that you asked, but let's look at what would be the mathematical better play for the house. Using my pai gow poker appendix 1 we get the following, assuming the dealer is banking:



By playing the hand as a straight the dealer can expect to win 96.46% of the money, as opposed to 87.49% as a full house. So, going by the straight rule is significantly better.

Personally, the way I code pai gow poker is to start with the highest ranked hand (five aces) and move my way up the page in an if/else if/else if/else if/end loop. In other words, I classify the hand according to the highest possible five-card hand that can be made and then follow those rules.

I think that this is how the house way is intended to be interpreted. As evidence why, consider the Canterbury Park house way. In particular the rules for flush hands, which reads as follows.

Six and seven card flushes play the highest two cards possible in the Low Hand.  
(Exception: If two pair are with the flush, the two pair rule will apply.) 

As with all other house ways, the lowest hands are listed first. If the hand was supposed to be played by the first set of rules encountered, then they would have followed the two pair rule anyway. There would be no reason to explicitly make an exception to follow the two pair rule, since it was listed before the flush rule.

For practical purposes, I asked two dealers how they would handle this hand. They both pretty much said that the written house way is just a guideline and that in the event of a confusing hand, ask the floor what to do, and then just do as you're told without asking why. One of the dealers said that an unspoken policy is to go by the rule that seems to fit that situation best. In the case of Foxwoods, there is a rule for straight hands with a three of a kind. Since that most specifically describes the hand in question, follow that rule, despite the fact that the full house rule covers it too.

To make a long story short, no casino house way I have ever seen, in either pai gow poker or pai gow tiles, clearly explains how to play every possible hand. They are full of contradicting rules. Until somebody cares to do anything about it, you'll have to go by the interpretation of whoever is working the floor. In my experience, these confusing cases seem to usually get ruled against the player.

I hope one of these days a player will get fed up with this and make a complaint to the appropriate Gaming authority the next time an ambiguous hand is set in a way that favors the dealer.

This question is raised and discussed in my forum at Wizard of Vegas.

I checked various Caesars properties on October 1 and found this pay table for the $5 progressive Face Up Pai Gow Poker bet.

• Seven-card straight flush = $3,922,620
• Five aces = $185,104
• Royal flush = $2,969
• Straight flush = $500
• Four of a kind = $375
• Full house = $20

I analyze this bet in my page on Face Up Pai Gow Poker. In this case, the Mega Jackpot, for a seven-card straight flush, is unusually high. I believe it to usually be in the low five-figure range. The bottom line is this $5 bet has an expected return of $9.55! That means an expected profit of $4.55.

You still have to make a negative bet on the base game. The minimum on that ranges from $15 to $100. At a house edge of 1.81%, here is the expected loss by bet amount on the base game.

• $15 bet = $0.27 loss
• $25 bet = $0.45 loss
• $50 bet = $0.90 loss
• $100 bet = $1.81 loss

At a base bet of $251.61 and $5 on the progressive, the combined game would have zero house edge.

However, two big reasons not to bet it are the 37% highest marginal federal income tax rate and the huge volatility. The average player who doesn't hit any of the top three progressives can expect to lose $3.38 a hand at a $15 base bet and $3.56 at $25. That adds up on an hourly basis.

Still, it is, mathematically, a very positive bet. At a $15 base bet, the combined player advantage is 21.39%. You don't see that every day.

This question is asked and discussed in my forum at Wizard of Vegas.