Ask The Wizard #252

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:

6 decks
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:

Stand -0.152739
Hit -0.000274
Double -0.004882

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.

In the four casinos in Colombo, Sri Lanka they have the following blackjack rules:

  • 6 decks
  • Dealer does not take a hole card
  • Player may "early" surrender, except against an ace
  • Dealer stands on soft 17
  • Player may double on any first two cards
  • Double after split allowed
  • Re-splitting aces allowed
  • If dealer gets a blackjack, player will lose only his original bet
  • Player may opt to win half his wager with any original five-card hand under 21


I show that combination of rules has a player advantage of 0.65%!

The last rule mentioned can also be found at the Pharaohs Palace in Macau, too. It is a breakeven game there, though, because of other bad rules, like doubling on 11 only. I have a strategy for the five-card half-win rule on my blackjack page at Wizard of Macau.

I heard that blackjack pioneer Ed Thorp also had a card counting strategy for beating baccarat. What do you know about it?

Tom from Hong Kong

I found two sources online that address your question. The first is a quote from an article I found:

But Edward Thorp and his computer are not done with Nevada yet. The classiest gambling game of all — just ask James Bond — is that enticing thing called baccarat, or chemin de fer. Its rules prevent a fast shuffle, and there is very little opportunity for hanky-panky. Thorp has now come up with a system to beat it, and the system seems to work. He has a baccarat team, and it is over $5,000 ahead. It has also been spotted and barred from play in two casinos. Could it be bye-bye to baccarat, too? —Sports Illustrated, January 13, 1964 issue

Thorp also addresses the vulnerability of baccarat to card counters in his book The Mathematics of Gambling. The link goes to a free online copy. Thorp concludes by saying:

Practical card counting strategies are at best marginal, and at best precarious, for they are easily eliminated by shuffling the deck with 26 cards remaining.

Interestingly, Thorp also says the tie bet pays 9 to 1. Perhaps that rule was more common in 1985, when the book was published. If memory serves me correctly, Binion’s paid 9 to 1 until the late 90’s.

My own analysis points to the same conclusion, although I studied the tie bet with an 8 to 1 win. I find the pair bets that some casinos now offer have the greatest vulnerability, but are still not a practical advantage play.

I asked Don Schlesinger about the apparent contradiction and Thorp’s baccarat team. Don said that he believed that Thorp did indeed have a team trying to exploit the tie bet. Either Thorp’s team found games with a cut deeper than 26 cards, or he had a change of opinion about it sometime between 1964, the date of the SI article, and 1985, when The Mathematics of Gambling was published.