Three Card Poker - FAQ
I noticed on your page for Three Card Poker you advise to stay with Q/6/4. A dealer in Tunica told me Q/J. Why the difference and is the Q/6/4 suggestion based on actual computer generated odds? Seems a pretty good game with low house odds, no skill involved. What do you think?
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.
How do you go about working out the separate probabilities for getting: (1) queen high, (2) king high, and (3) ace high in three card poker?
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.
How is the house edge derived in Three Card Poker (ante/play bet)? I haven't played yet, but my guess is that the edge comes from the fact that the player must decide whether to fold before the dealer. If the player folds and the dealer does not qualify, does the player get paid on the ante anyway? Otherwise, it would seem an even game (which is obviously not the case). Thanks in advance.
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.
First, let me congratulate you on your great website. I tell everyone I know that if they are going to gamble to make sure that they visit your site first! My question is on Three Card Poker ante & play. If you know one of the dealer's three cards how should you change your basic strategy and could you obtain an advantage over the house and by how much?
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!
In Three Card Poker would it be prudent to increase your bet after say 5 or 6 losses? I know the Martingale system is bad news but since in Three Card Poker you have bonus payouts for better hands I thought it might be worth a chance. Please think about this for a minute before you answer.
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.
I was playing 3 card poker at the Venetian this weekend and a friend of mine drew 3 Queens in the same suit in 2 consecutively dealt hands. Was very curious as to what the odds are of that happening?
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.
What is the probability of getting two identical straight flushes (in both ranks and suit) two hands in a row in Three Card Poker?
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.
Wizard, I have just read an article on another site that was talking about Three Card Poker. The author called the "Pair Plus " bet a sucker bet ,compared to the "ante" bet. Of the two bets , I thought the "ante" bet was more of a sucker bet because of its higher house edge. Also, what is the element of risk in Three Card Poker , compared to Let it Ride , assuming the player is starting with $5 bets in the three betting circles for Let It Ride, and $5 in the Pair plus and ante bet in Three Card Poker?
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.
Wizard -What is the advantage in Three Card Poker in playing two hands versus one, if there is one? Some casinos allow you to play two, playing the first hand before looking at and making a decision on the second hand. Some casinos will only let you play the second hand blind, which I’m sure is no advantage to the player. Thank you
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%.
In Three Card Poker which is the better bet, the Pairplus or the Ante & Play?
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.
When playing Three Card Poker what is the optimal ratio of the Pairplus bet to the ante/play bet? (Since if you have a Q/6/4 or better but no-pair and the dealer does not qualify - you lose the Pairplus bet but win the Ante. In this case a equal bet on both with be a push - but doubling the Pairplus bet on the Ante - would give you a win of the Pairplus bet amount)
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.
In your Table of probabilities in Three Card Poker you state the number of combinations as 9720 for Queen to Ace high and Jack high or less as 6720. I’m trying to calculate these probabilities myself and have failed to do so. I would be obliged if I you could let me see your calculations. Much obliged.
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.
Why is a straight a higher hand than a flush in the new casino game 3 card poker?
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.
Playing 3 Card Poker and using a Shufflemaster, I received 2 consecutive hands with the same cards, same suit (from the same deck). I was sitting at first base, so these were the first cards out both times. What are the odds of this occurring on consecutive hands?
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.
What are the odds of getting the same hand as the dealer in Three Card Poker, and how did you got your answer.
The probability is about 1 in 1 in 903.76 but the solution is too involved to explain.
What is the optimum ratio between the Pairplus and Ante bet in Three Card Poker?
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.
Good Day. May I say that I have thoroughly enjoyed looking through your web site. Can the house advantage on any particular game be affected by the Limit/Maximum being too high (i.e. over $5000 on the pair plus), especially with three card poker? Best regards.
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.
Is there an optimal strategy for the amount of the Pair Plus bet in relationship to the Ante? (i.e. should the Pair Plus bet be 2x, 5x or some other increment of the ante to maximize potential return?)
Bet 100% on the Ante, because the ante has the lower element of risk.
Hi Wizard - thanks for a great site! The basic Three Card Poker strategy is to play Q-6-4 or better to minimize the house edge. But what is the best strategy to minimize the element of risk (given ’5-4-1’ rules)? Your site reads that playing Q-6-3 loses 1.00255 units; but then betting that hand will dilute the element of risk, since the bet is suffering a house advantage of only 0.255% (which is less than the 2.01% 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.
First off, I love your site. I occasionally play Three Card Poker at Grand Victoria Casino in Elgin, IL. They have a Pairs Plus Pay Table that isn’t listed in your Three Card Section:
Three of a Kind 30:1
Straight Flush 50:1
Thanks. This pay table has a house edge of 5.10%.
I’m a croupier working in a UK casino and just thought I’d point out that our version of Three Card Poker has a slightly different paytable than any of the ones listed on your page.. We pay 35-1 for a straight flush, and 33-1 for three of a kind (then 6-1, 4-1 and 1-1 for the other 3).
The house edge of that pay table is 2.70%.
What is the standard deviation on the Ante & Play bets in Three Card Poker?
I have reviewed your site and am looking for casinos in Nevada and Mississippi that have the best pay out schedules for each table game (I prefer 3 card poker). Do you have a listing or link that shows the "loosest table games" casinos?
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 you have a sloppy Three Card Poker dealer who shows one of their cards, how does this change the house advantage? For instance, you are holding a Q-8-6 and you see that the dealer is holding a Queen or higher, the player would most likely fold the hand thereby affecting basic strategy and the house advantage.
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.
In a Three Card Poker tournament with $1,000 in chips and only 30 hands being dealt, what would be the optimal bet sizes for the Ante and Pair-Plus?
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.
I play three card, Caribbean stud, and four card poker at machine shuffled tables. I am amazed the number of times a playable three-card hand frequently is dealt in a four card game, and a playable four-card hand is dealt in a Caribbean stud game. It makes me wonder if those shuffling machines aren’t pre-programmed to the house’s advantaged. Are these machines really random or are they programmed for the house, and if programmed, isn’t that illegal?
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.
Last night, a fellow player at a 3-Card Poker table dropped his cards on the floor. After the cards were dealt, he had picked them up and was opening them up with one hand while showing them to his son who was a "spectator" behind him. Somehow he fumbled them and all 3 fell to the floor. I had already looked at my two hands ( a flush and a straight!) and placed my bets. The floorman told everyone all their hands were "dead". When I asked for the supervisor, he came over, thought about it for a few seconds, and also agreed it was a "dead" hand. Is this the standard rule on this type of occurance? Should all hands have been disqualified, or just the offender’s (who, by the way, had a Q,6,5). I was betting $10 per spot, so my loss was $150 in winnings (I am assuming that I beat the dealer - he never did show his cards). Should I have asked for a copy of the written rule on this? Should I have not given in after the supervisor’s ruling?
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.
I deal 3 Card Poker and I like to play my tip bets on the ante. Will I win more money for myself on the ante or should I be going for the long shot on the pairs plus bet? Thanks!
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.
Hi, Wizard. The casino here allows players to take over another player’s hand in Three Card Poker if he wishes to fold. The player taking over the hand must put up the play bet himself. Should I take over his hand if I know that one of the dealer’s card is 2 to J (low rank card)? What is the player edge? Thanks for your reply.
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.
Three Card Poker toking: If a player tokes on the ANTE bet, but does not toke on the dealer’s play bet, then the toke bet is technically "surrendered," - just as if the player had refused to make his own PLAY bet to play out the hand, technically. However, most dealers ignore the absence of a matching PLAY dealer’s bet, and are loath to petition a generous player for fear of appearing selfish, so just pay the ANTE dealer’s bet. Of course, the PLAYER must ALWAYS make a PLAY bet to match his ANTE bet if he wants to play out his Three Card Poker hand. Is there an unfair dealer’s toke edge to this practice, or is it similar to surrendering after double down for less? I think it’s unimportant, as since the player’s hand is active, then the toke is active, and the risk versus reward are simply cut in half for both the dealer and the house, - no harm done. The Wizard’s view would be GREATLY appreciated by this dealer.
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.
I would like to know the decisions per hour in particular for the poker games: Three Card Poker, Caribbean Stud, and Let It Ride. Thank you.
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%|
As a dealer, I can help with the Three Card Poker positioning of the toke bet question. A smart dealer will position the bet between the pair plus and ante spots and then can take advantage of either bet winning. Shhhhhhhhhh, don’t tell on me.
Anonymous from Niagara Falls
Don’t worry, I won’t give your name.
Great site! A must see for Vegas newbies and those thinking of striking it rich through gambling. Is there an optimal ratio to wager between the ante and the bonus bet in the Texas Hold’em Bonus table game? Is it simply 10% bonus to ante since you calculated 90.4% of bonus bets are losers?
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.
Why does the dealer have an advantage in Three Card Poker? First, it is three random cards vs. three random cards. Second, even if the dealer beats you, he has to qualify, so wouldn’t that give YOU the player the advantage? I know, it’s stupid but my brain can’t comprehend.
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.
I went to Vegas last month and played Three Card Poker for the first time. I got a straight flush, and was so excited to win that I didn’t notice that the dealer only paid out 20 to 1 instead of 40 to 1. I lost a few hands and left the table to go cash in, then realized what had happened. My question is, if I notice an error in the future, what should I do? I’m assuming after I left the table it was way too late, but what is the rule if I’m still at the table? If I don’t point out an error before the next hand begins, is it too late?
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.
When playing Three Card Poker, and spotting a dealer card, I am aware the edge is 3.48%, played properly. However, the game I play has the 1, 3, 4 Ante Bonus table. I am curious as to the effect this has on my edge over the game, as well as my edge if I only identify ace/paint/non-paint.
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.
With regards to your California Three Card Poker section, there is a casino in Oceanside that offers the game with similar rules. The casino shows one card, but in order to win your ante bet the dealer hand does not have to beat the player’s hand. What is the house edge in this game?
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.
When using a five dollar match play coupon in Three Card Poker, would the optimal strategy change on the Ante/Play portion, based on the following assumptions? The $5 match play must be placed only on the ante bet. Also, in Washington State, the match play coupon has "no monetary value," and thus is not counted when determining how much must be bet on the play portion. Thus, a $5 bet on the ante with a match play requires a $5 bet on the play portion, not a $10 bet. Thank you.
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
On your site you show the difference in house advantage for Three Card Poker with the Mini-Royal payout on the Pair Plus bet. At a local Casino here in Arizona the Mini-Royal plays on the Ante Bonus instead of the Pair Plus. The Ante Bonus pays as follows, 3 of a kind 6 to 1, straight flush 8 to 1, mini royal 50 to 1. What is the change in House Advantage on the Ante and Play with this pay table? Thanks,
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%.
You mentioned on your Wizard of Vegas site that dealers often incorrectly don’t pay the Ante bonus in Three Card Poker when the dealer beats the player. What do you think this error costs Nevada players on an annual basis?
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.
Your blackjack basic strategy tables are based on maximizing the expected value per hand. However, are there situations where doubling or splitting is such a marginally bad play compared to hitting or standing that the cost of the error is less than the house edge of playing an additional hand?
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.